When should one give up on others?

Depends on what it is. Most people do not change and do not want to change, or they say they do, but they don't make the effort because it takes work to change. You can't make anyone change and most people actually do not want help at all -- they want someone to vent to and that's it.

You can make suggestions, offer to lend a hand but if it is not accepted, then let them go.

It's true we all need someone some times, but when is enough, enough?

Reason I ask,

I recently tried to help another in life, but found me running into a brick wall. Person is out in the dumps with little family or support. I gave them my hand and they tossed it back in my face. I can see the potential in anyone, but... I don't have the question to ask correctly, as my question was answered, but I would like to take the time to listen when others feel it is enough

You should give up trying to change them after one or two tries. But feel free to hope they come around in the future- as long as you are able. A lifetime, if possible.

Usually after half an hour on Yahoo Answers

When you can't get through to them, when they let you down enough times, when they show you disrespect. And when you may as well be talking to a brick wall.

Whenever they let you down a couple of times

I have already given up hope on humanity -_-....