I used to have a lot of trouble with this, but I live near downtown, so I do a lot of walking and a lot of crossing. You get used to it.
It's good to be a little nervous when you're going to cross, because you're a soft fleshy person treading into the territory of two ton death machines. That's what you have to pay attention to. Watch the signals, look both ways, etc. The people driving them, not so much.
People are not staring at you. They're generally paying more attention to their stereo, checking themselves in the mirror, talking on the phone (grr!), thinking about what they're going to do when they get to their destination, etc. You're just a momentary blip on their radar. There one minute, gone the next, never to be thought about again.
If a driver gives you shit about doing something entirely normal like using a crosswalk or making sure they see you before crossing at a stop sign, give them the finger. They are assholes of the worst sort, and they deserve it. Their rage has nothing to do with you and everything to do with them being forced to think, for one second, about something other than themselves.