Too short, and too simplistic. The real answer is: it depends.
If the child will be flying as a free lap child on a US domestic flight, the answer is usually yes, not for identification, but for proof that the child is indeed under age 2 and eligible to fly free. A photocopy of the birth certificate will normally suffice for this purpose, as long as the adults he is flying with are his parents, so that the names match. If you are flying with a grandchild, etc., it is better to bring the real birth certificate, as it might be questioned whether or not you just copied one at random.
If the child will be flying on a US domestic flight in a paid-for seat, no identification is necessary.
If the child will be flying into the US on an international flight, he/she needs a valid passport.