Children 3-16 on domestic flights will not be asked. Technically they don't need it until age 18, but once they start looking like they might be old enough to drive they are often asked for ID.
Lap-babies under age 2 may need proof of age when flying domestically. For really tiny infants you won't be asked, but once a child starts walking it is quite likely that the airline will check.
On any flight where you are crossing the US border, the child will need his/her own valid passport, regardless of age.