I was just researching this same question. When we flew in March, the airline asked for ID for the children. I didn't have any as we have never been asked for it before. She said that the TSA may ask for it, so be aware and to be sure we have it next time we flew.
I can find one child's birth certificate, but not the other and we leave in 2 days (yes, should have done this earlier). I have printed out the TSA rules, so hopefully it won't be a problem. I can't figure out why we were told to bring it next time when the TSA website clearly states that ID is required for adult passengers only.