We fly all the time mostly on United, and get asked for ID for ds about 50% of the time. If you don't have ID, when they ask, you don't get on the plane. So for sure make sure you have it in case they ask.
Its not just to varify age for under 2's, its also to varify that you are allowed to take that child out of state or whatever, if you have their ID, you are much more likely to have permission to take them on a trip.
You can use a birth certificate. You can also get an ID from the DMV, it looks like a drivers licence kind of but is just an ID, nice to have if you travel a lot because its easier to carry (I tend to loose or rip paperwork in my bag, cards just live in my wallet)