We just got back from a trip that included our two nephews - ages 5 & 1. Never at any place were their parents asked for any ID for them.
As long as children are traveling domestically with their parents, there should not be any need to have ID for them. The only time I have heard that problems may occur is if a child is traveling with only one parent and the destination is outside of the U.S. (due to custody disputes/kidnapping).
Some people do like to carry a birth certificate or passport for their children, but it isn't necessary. If you have any doubt, check with your airline to see their policy on this. Most post it on their web sites.