A lot of times, support is overlooked in public games, due to a disorganized style of play that lends itself more to carries. This doesn't mean that support is a poor choice when playing a more organized game, and even a 2-man arranged team can carry more effectively when one person chooses to babysit.

First off, a bit of an explanation. Babysitting is a term that comes from DotA, like many other terms used in LoL. It refers to a champion who follows around and assist a specific hero, almost always a carry.

Second, there are 4 champions who babysit exceptionally well. Not all champions listed as support by Riot are babysitters. These heroes are, in no particular order:
Kayle
Alistar
Janna
Soraka
Kayle is easily the most popular, and Janna and Soraka are the least played of the four, and often treated as useless. I expect Soraka to become more popular if/when it becomes possible for mages to carry, and when it becomes possible to see an opponents picks.

Third, some jobs a carry must perform.
a) You must prevent the hero you are to protect from dying.
b) You should create opportunities for your carry to kill opponents.
c) You should help your team, especially your carry, escape from ganks and situations where an offensive move turns bad.

Section 4 will be a guide on how to babysit with each hero, starting in the next post.

Kayle
Kayle is the quintessential babysitter, and one of the best if not the best.

Reckoning
Kayle can slow an attacker's move and attack speed, which is both offensive and defensive. Reckoning can be used to harass, to slow a pursuer, or to slow an escaping foe. This is how you set up your hero for kills. REMEMBER NOT TO USE THIS ON A LOW HEALTH OPPONENT UNLESS THEY ARE GETTING AWAY. Kayle does not need the money as much as a carry does.

Divine Blessing
Kayle can also heal, and increase the movement speed of an allied champion. This helps your carry escape, keeps them fresh after harassment, and can be used to speed them up so that they get a kill. Dont forget that you can use this on an ally or yourself in order to make sure that they/you get to whatever location they need to. One of Kayle's best advantages as a support hero is that she is fast and can make her allies fast (due to blessing).

Righteous Fury
It is unwise to level this skill very far when babysitting. The splash may be nice, but you shouldnt farm creeps, because the carry needs money. Other than that, it's a slight boost of damage and the ability to stay a safe range away from enemies when attacking a tower. Don't max this early, get at most one level of it before level 13.

Intervention
This is the reason that people love playing on a team with kayle. On an organized team, this is license to dive into a dangerous area for a carry. On an unorganized team, it's the saving grace for people about to die. It has a huge cast range, so you may be able to rush your way up to a fleeing near-dead teammate and save them with it, even if you're coming from the store. This is best used to make a fragile carry like Twitch, Ashe, or Master Yi able to enter a teamfight and kill the enemy.

Other points of interest on Kayle.
Good items:
Boots of swiftness. A babysitter needs to be there for their carry all the time.
(if the other team is balanced heavily towards casters or physical dps, mercury's or ninja tabi are acceptable. Ideally, you awnt to be able to grab swiftness, though.) DO NOT GET BERSERKER'S GREAVES.

Stark's Fervor. You yourself do not need the lifesteal or attack speed, but putting the gold you earn from assists into giving your team another reason to hang with you and freeing up one of their item slots is good.

Soul Shroud. You can use everything this item gives and your allies can use the mana and cooldown reduction.

Aegis of the legion. protect your carry and everyone else around you, and get health.
Survivability items, specifically a banshee's veil, a Sunfire Cape, a Force of Nature, or a Warmog's. These should be taken after you complete aura items.

SUMMONER SPELLS
Cleanse. This is a must on almost every babysitter. You need to be able to get out of silence, stun, or anything else that prevents you from safeguarding your baby. I would never put anything else in its place.

Rally. Kayle's second spot is really open to debate (bar a few summoner spells), but being able to increase your carry's damage and having a group heal is nice.

Teleport. Be where your team is, be where your carry is. Nuff said.

Clairvoyance. It's good to be able to scout out where the other team is, and to free up a summoner spell slot for someone else who might otherwise pick clarity. This is best used on an organized team

Exhaust. This accentuates the ability of a carry and babysitter to go off as soon as possible, and grants you another way to defend your carry from attack.

Flash. You should attempt to survive as well, this is a good way to get out of ganks after you cleanse.

Mistakes people make:
Buying Atma's impaler(s) and health items. Your job is not to do damage, and you arent the largest tank in the game. Focus more on team oriented items like the aura-givers.

Going DPS: this is completely wrong. Kayle was not meant to carry a team, and is never going to be the damage dealer in a good match.

For now, that's all. Suggestions are nice, and i'll post Alistar's analysis next.

Alright, a couple posts I'm going to mention. PosPosPos mentioned Janna being underrated, which i think is largely right, at least as a babysitter. This struck a cord with me, so I'm doing Janna instead of Alistar next, thought this doesn't mean that Janna is better.

Another poster mentioned Taric. Taric can babysit, this is true. But he lacks the ability to set people up for kills in the way the aforementioned heroes do. Soraka doesnt set people up for kills like the other 3, but she serves as a babysitter in a unique way, I'll go over her last. There will probably be other heroes people would like to talk about, but in my experience, these 4 babysit the best. Soraka is unique, remember, she'll come last.

Janna
Janna is commonly regarded as a bit of a chump hero. She is unlikely to output a high kill count, as are other support heroes, but this is not why she is regarded as bad. Most people build Janna for AP. This is very wrong. The majority of your usefulness will not be based on your ability to deal damage, and even though it does buff your heal and shield, there are better ways to play.

Going trough her skills...

First, her passive is mildly nice for your team. It is unlikely that this will make a huge difference, but every bit of move speed counts. With Kayle, the passive was sort of a non-issue, because kayle shouldnt be attacking enemy champions very frequently anyways, and allies shouldnt be dying around her.

Howling Gale (The Tornado)
This move is misunderstood, especially because it has potential to make a huge impact on the early game and escapes. First off, don't push your luck trying to channel the spell, and try to cast it from the bushes. If it appears that they'll discover the tornado, you should send it off early too get some harassment in for the carry. Second, don't fire the tornado directly into the creep wave, with 2 exceptions. The first is when you are pushinjg a tower and want to delay and damage the creep wave as much as possible. The second is when a fresh wave is starting and your carry can scoop up low -health creep kills for a lot of fast gold. Janna can speed up a carry's farming ability very quickly this way. In situations other than those, attempt to hit enemy heroes without touching the creeps. You're great at harassing. The tornado is also great at helping a chase or escape. In situations where a champion is escaping, you should aim it at where they will be, and then quickly toggle it on and off. "Flicking" a tornado at them is done far too seldomly. This is also the policy for escapes. Simply use the tornado as soon as it begins channeling without waiting for damage to increase. Even ganks and 2v2 skirmishes are best handled by flicking a tornado if you don't have one ready; play it by ear to see if HG is needed as a nuke or a disable. If it appears that the tornado would steal the kill, let it channel and go off after the enemy champ dies. You don't need the money as much as your carry.

Zephyr
This is what gives janna a definite edge as a babysitter. She, more than any other babysitter, will always be able to follow the carry. She gets a movement speed increase and can float through units. It's nice to have. In addition to this, you canm shoot the zephyr at an enemy to slow them. This is for chasing down excapees, don't use it unnecessarily. If you do, you'll be slower. Make sure not to KS with the zephyr. Other than that, straightforward.

Eye of the Storm
Use this on towers that are being attacked and friendly heroes that are being attacked. Try to make sure you don't need it yourself. Use it on a champion that is about to do something risky, like a tower dive. Simple. A good spell.

Monsoon
This spell turns a lot of people off to Janna, because soraka can heal all allies for more. This doesn't mean the spell is useless, though. Janna can use this spell defensively, or interestingly enough, offensively. The push can be used early game to separate a champion in a sort of 1-2-3 spell combo. You flick a tornado at someone, slow them with the zephy, an when behind them, you use the monsoon. This can scoop your carry up a kill easily, considering a slowed, damaged, and separated champion fell into their lap. The defensive aspect is simple. After a teamfight, if your team is damaged, you can heal them all. You can save a person who is being assaulted by pushing everyone back and then shielding them. You can heal your carry early on if you're playing against offensive opponents, or the carry is damaged for whatever reason. Other than that, it's a very good amount of healing, and if you keep it channeled for the full time, your allies will love you for it.

Items:
Boots of speed: once again, boots of speed are best, because even if you'd like the extra survivability or magic penetration, you should try to always be present for the carry. These facilitate that. If things get REALLY bad, you can reach for mercury's treads or ninja tabi, but try to avoid this. Do not get sorceror's shoes.

Soul shroud: Health, mana, cooldown reduction. Janna likes all of these, and it makes it easier for her to play with a mana-hungry carry like twitch, or a mana hungry support, like blitzcrank.

Aegis of the Legion: Janna is a bit squishy. This helps. It also helps everyone else.

Banshee's Veil: Janna is squishy, and dodging a spell really helps her. I'll talk about rod of ages in a second.

Rylai's Crystal Scepter: Janna can use health, and her allies will thank her for the slow. Don't rush this, it's a luxury item for times when you do very well.

Frozen Heart: This is the last Item I'll advise you on. If you can afford every item on the list before this item, and this item, you are most likely going to win anyways. Your sixth item could be a sword of the divine for all it matters. This is just a boost to survivability late game and a nice dose of cooldown reduction.

I actually DON'T recommend getting a rod of ages, at least not early. You can substitute it for a rylai's scepter. While it does give you the potential to be tanky, a banshee's veil is more along the lines of what you need. Your AP is not very crucial. Mana will come from the soul shroud and the veil. Health will come from every item other than boots and the heart.

If you find that you're consuming mana too quickly early on trying to build this, you can grab a meki pendant and make a chalice.

DO NOT GET A RAGEBLADE. YES THIS HAPPENS.
DO NOT GET ANYTHIN THAT HAS TO DO WITH YOU ATTACKING THEM PHYSICALLY. YES THIS HAPPENS.

Summoner Spells:
Cleanse: Once again, you should be ready to defend your carry or support them at all times. Make sure that fiddlesticks/kassadin/whatever stunners the other team has don't stop you. This should always be a pick.

Exhaust: Same reasons as with kayle. You want to make sure your carry gets kills ASAP.

Clairvoyance: Mapa awareness is good for your team, make sure you only grab this on an organized team. Solo queuers will not benefit much.

Rally: More healing and damage for your allies.

Flash: Gank evasion after a cleanse or catching up to someone to slow them down for the carry.

Mistakes People Make:
AP janna: She does very poor damage. Made with only two nukes, she was clearly not intended to be a killer. You will do poorly and your team will be dragging you behind them. This build will usually also get sorceror's boots, which make you slower than boots of swiftness.

DPS Janna: I don't know why, but i've seen this 4+ times when i was tooling around with a friend who was new to the game. This has even less logic behind it than AP janna. None of the items are recommended, so people clearly think this is good idea, and arent just picking items because the game told them to.

Certainly. Fiddlesticks is clearly not without his merits as both a harasser and a disabler. Yet he does not really babysit a carry.
His best harassment comes from dark wind, which you can't be sure of the damage output of. This can lead to KSing a carry, which is bad. This isn't the pressing issue, though.

You'll notice that each of the heroes i listed (bar soraka, she's a different case) can defend their baby against multiple assailants. Janna and alistar are great at making sure that a hero escapes, and in addition to this, require only a small investment in their disabling skills to do so. Fiddlesticks has a single-target 3 second disable and a silence. This is good vs. mages, (but soraka is preferable to fiddly in a mage lane), but lackluster against a lane with champs like teemo, master yi, and ashe. In addition to this, once the carry is hurt, they're hurt, and fiddlesticks can't heal them. Fiddlesticks is also woefully slow, even slower than alistar.

When deciding which champions are babysitters, look at their skills, and think about which skills are geared towards helping other champions. Fiddlesticks has 2 skills (drain and crowstorm) which can't really be said to benefit someone other than fiddlesticks more than him. Janna, Soraka, and Alistair have no skills that benefit other champions less than them, and kayle has only one- righteous fury.

This thread is not supposed to be inclusive of each USEFUL support hero, if it was fiddlesticks and blitzcrank would most definitely be in it. It is more to the tune of heroes geared particularly well towards keeping their baby carry alive, fed, and happy.

I'm probably going to have to add taric onto this thread eventually, though it's a peeve of mine to play him. He can heal, he has a built in rally, and he can help out a physical carry quite a bit. I'll tack him on later.

what kinda runes? would you suggest with kayle