As others have said it all depends on how big said kid is, his maturity, strength and his experience from dirt bikes, quads or anything that might cross over to a snowmobile. I started at age 2 on a kitty cat then at 6 up to a 340 then at 10 or so to a 440 fan then at 12 to liquid cooled machines. Granted I was small at those ages but I had been riding dirt bikes from age 2 as well and had a jet ski growing up as we lived right on the Kankakee River near Joliet, IL. Plus it helps when said parent is involved and takes the time to teach them right from wrong and how to properly ride safely no matter what kind of machine or powersports machine you are on. And it helps if someone can also help teach them how to properly maintain what you own as it will save you thousands in repair bills. Dad had me rebuilding chevy 350s at age 8 so he hammered into my brain that preventative maintenance is always the best thing for your machines and will save you $$$ in the long run.
Also I might suggest a DNR or some other professional training course. Many states require that you have to be a certain age before being able to ride now and must have said training certificate on their being or machine at all times along with a proper ID even if it is a state issued regular id if they are not yet 16 and have a driver's license.
Good luck with it!