I brought a copy of B's birth certificate and the agent at the check luggage window (I forget the term for this) asked to see it but I've read that Southwest is the only airline that does this. No one else asked for ID.
Southwest will ask, always. I have never had any other domestic airlines ask, even with a 23 month old. We would always bring her passport or a copy of her birth certificate.
^^^what fivedogs said^^^. I flew exclusively with DD as a lap child before she turned 2 and was often asked for her birth certificate on different airlines.