Summary: Some practical suggestions for parents dealing with stage fright in a young performer in a music-education setting.
The rush of adrenalin that often accompanies public performance is not necessarily a bad thing. For many performers, a minor case of "nerves" helps them concentrate on the task at hand, blocking out distractions and leading to a performance that is more lively and more proficient than most of their practice sessions. The term stage fright is usually reserved for cases in which fear causes the musician to give a performance that is noticeably worse than they give in a practice, rehearsal, or teaching setting.
Very, very few human fears are in-born. Fear of falling is an example of a fear that apparently does not need to be learned. The vast majority of fears, though, are learned responses. They are either learned from others (you are afraid of spiders because when you were three you saw someone react to a spider with fear), or learned from personal experience. So the good news is, there is no natural, innate tendency to fear performing in front of people.
The bad news is that, since fear can be useful for survival, it is a response that is learned very quickly and easily. A single bad experience is enough to cause a person to learn to fear a certain kind of situation. Since fear is much harder to unlearn than to learn, this is one situation where "an ounce of prevention" is much more useful than several pouinds of books on "overcoming your fears". So, if stage fright is not an issue with your child, your main concern is to see that it does not become one. Here are some helpful do's and don'ts:
Encouragement based on truth is a major weapon in both preventing and curing stage fright. The "based on truth" part is important. If the statement "that was fantastic" or "you were really better than the girl who won" isn't true, don't say it! On some level, your child is probably aware of the truth (and if not, protection from the truth can slow or stop progress), and will not appreciate falseness, no matter how well-meant.
Useful encouragement is usually based on specific, true, positive statements, such as "you are so much better than you were last year"; "that was a lively performance"; "you're doing great for someone who's only been playing for one year"; "that's the best I've heard you play that difficult piece"; "I was amazed at how fast you could play the last movement"; "the slow part was very pretty"; or even "hey, you hit a lot more high notes than you missed". If you know enough to give even more specific positive comments ("you nailed the high B flat!"; "you've got those tricky 6/8 rhythms down now, don't you?"; "your slurs are so much cleaner than they used to be"), do it.
The "fight-or-flight" adrenalin response that fear causes is not a problem when a performance requires an all-out physical effort, as it does in many (but not all) sports situations. The physical reactions that accompany the fear response can be a big problem, however, for anyone trying to give a highly controlled mental, and physical, and emotional performance, and there are some effects that prove particularly difficult for musicians.
It should be obvious from the above list that severe performance anxiety should be avoided if at all possible, and that a child who has just experienced it should be treated with sympathy, not judgment. What to do to help a child who is having this problem?
Preparation is the single best hedge against performance anxiety. Like a rescue worker who has been well-trained to do certain things in dangerous situations, a musician who is extremely well-prepared for a performance may find that most of the required actions happen automatically even during a panic. Practicing the music is crucial. Practicing specific things (like taking a deep breath before playing or fingering difficult passages in a certain way) can also help make necessary things happen even when clear thinking is not possible. Practicing in stressful, distracting environments (for example, in front of the family in the kitchen while dinner is being prepared) can also be helpful. Performing often in unstressful situations can also help calm a child who has developed performance anxiety. Deciding and practicing every single aspect of the performance (announcing the piece if required, where to stand, how to bow, etc.) is helpful both in lowering anxiety levels and in causing a smooth performance during a panic. Some students may find it helpful to identify key words or phrases ("big breath"; "one-and-two-and") to write in the music or say to themselves at a specific spot in the performance. They can then practice responding to that phrase in the correct way, until the unthinking physical response is automatic if they see or remember that phrase.
If stage fright continues to be a problem even when your child is well-prepared, it may be necessary to deal directly with the anxiety. There are many professionals who specialize in helping people overcome performance anxiety, as well as an enormous amount of self-help material out there. Obviously, all of the popular approaches have helped many people, but the best approach for each person depends on inividual personality traits and learning styles. You may have to try more than one approach, and you should feel free to alter a useful approach to better suit your child. Enlist as many people as you can without embarassing your child, to help in the anti-anxiety project: music teacher, director, bandmates, and friends may all be sympathetic and helpful.