I believe it depends a great deal on the personality and ability of the child involved. On average, I would say 10 or 12 is a reasonable age range for a beginning hunter under supervision by a competent adult. My own rule is that two factors need to be present before a youngster should be allowed to actually hunt big game:
1. Proper training by a competent adult in firearms safety and use.
2. The ability to shoot a rifle accurately that is of sufficient caliber to obtain clean kills on the game hunted. One should not be out hunting mulies or moose with a .243 in my humble opinion.
Yes, i know, kills can be obtained with rifles that small, but there is not sufficient margin for error for a kid.
For small game, like rabbits and squirrels, that may be hunted with small caliber weapons like a .22, an younger age may be appropriate, depending on the kid involved.