It's probably taboo to mention that many professional musicians regularly visit their doctor and get pills (beta blockers) to suppress adrenaline production.
This allegedly doesn't make you less nervous but it avoids the physical symptoms like shaking hands.
Note that despite suffering from stage fright myself, I don't use or endorse this approach (although I have to admit I'm somewhat curious about it).
The core of the problem, as others before me have pointed out, is our wrong "model of the world", in which we attach too much
importance to our performance (or rather: mistakes we make during said performance, and the reaction of an audience to such mistakes).
I have yet to find a good way to change my model of the world though ;)