there are many ways.

one: stuns. you can easily stun a yi or shaco or w/e and then kill them in 2-3 seconds.
two: rammus. Rammus is the anti-carry, and with thornmail and sunfires, can literally taunt them and they will die.
three: oracles/wards. once assassins lose the element of surprise, they become quite weak. Try blitz-pulling a shaco out of stealth. you will literally lol him to death.

typically the best way to counter a smart carry is to not let him get powerful in the first place. This means do not feed the carry, do not let the carry successfully set up ganks, do not give the carry level advantage. if this means exerting a significant amount of your resources in shutting down carries, it will be worth it late game. remember, carries excel late game but are pretty weak early game. capitalize on this. I'll give you a hint, casters are powerful early game but are weaker end game. try to use them to prevent the carry from getting a huge end game advantage by shutting them down early game using said casters. Annie, Veigar, Karthus are all good for this.

I main Rammus, and I love hunting out the carries, taunting them with a thornmail on, and watching them explode. Thornmail recently got buffed, so I can notice the carries hp go down even faster.

Taunt is getting nerfed to work with cleanse as well as mercury treads this patch, so not sure how good that is going to work out.

The trick to the anti-carry is burst dmg and a stun. Failing that, executioners calling if very powerful in reducing their lifesteal.

jax use to be THE melee anticarry, as people have said stun and burst damage, he has good enough burst damage and dodge over 50% melee/ranged attacks if built right. and that directly results in stuns. also people tend to underestimate his dps and i love to see when a yi thinks he can take jax down and gets shut down completely and die/run away.
but now there's sword of the divine, so i guess he's useless now.