Depends on the child.
I got a Red Ryder for Christmas when I was in the 1st grade, so five, but shortly thereafter lost it for taking pot shots at my sister's barbies in the house. (Never sure whether it was taken for shooting in the house or target selection)
I started shooting in the backyard with Pops when I was about seven, and got "my" first rifle at 11 in the sixth grade. I took to shotgun shooting, low brass 20 ga field loads and then low brass 12 ga loads the next year.
My older sister, however, didn't get real invovled with shooting until just recently, having never been more than a handful of times before Christmas and now evovled to the point of exhausting my supply of small pistol primers in a matter of weeks.
A lot of parents seem content, in my experience, to let their 11 year olds handle a gun for the first time during the rifle merit badge at summer camp, which I think is a great enviroment for it.
NRA Life Member
Big Sister: "You should be sponsored by Allen"
Me: "If you can't shoot good, at least look good walking to the firing line."
Big Sister: "Can you not afford a Pelican? Then buy an Allen gun bag."