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 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.
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.
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.
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.
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.