I think "stage fright" can mean different things. For some, like
@Stumbo was alluding to, we're talking a type of anxiety disorder that may warrant therapy and medication. I have a daughter with a social anxiety disorder and it's WAY different from someone who's merely shy.
Probably more common, though, is simply natural nervousness of being looked at by crowds of people and having to execute a performance where there's a risk of making mistakes. The best ways to deal with that, as others have mentioned:
1) overprepare. However tight you get in rehearsal, you're going to lose 15% when you go live. So get to the point in rehearsal that you know your parts backwards and forwards and can play them in your sleep. Muscle memory does a lot to kick in during a performance and carry you through. I think it was Suzuki who said that amateurs practice until they get it right, but pros practice until they can't get it wrong.
2) live in the moment. You WILL make mistakes. The beauty of live performance is that the instant you make them, they disappear, gone into the ether. All that matters is the NEXT note. Don't linger over mistakes.
3) Fourth wall. This is an actor's technique. Build an imaginary wall between you and the audience. Actually visualize what's on that wall - a painting here, a TV set in the corner, a fireplace, whatever. Spend time over it and make it vivid. When you look out in the audience's direction, don't look at them, look at your wall.
4) breathe, and channel the nerves into energy.
By the way, in my experience big crowds are way less nerve wracking than small ones. If I'm seeing a handful of specific individuals I know, I start getting distracted by worrying about what they're thinking and if they're enjoying themselves. But a crowd is just a mass of anonymous faces.