I work for the airlines and no you your children do not need a photo ID... however starting in Decmeber anyone traveling outside the US will need a passport (some places right now only require a birth cert/ social sec card/photo Id -- but eventually will require passport only )
At the airport, they just check my childs ticket, then take a look at them and make sure a ticket matches a person.....and I believe on the ticket it states CHILD so...that's about all they need. No id required.
Traveling in the US, we have never had to provide ID for our DDs.
In fact, my oldest DD (who is over 18) was kind of upset that they kept telling her they didn't need to see her ID.
I have taken many flights with my children and have never needed anything.
With that said we are going down with my nephew and he will be a lap baby and SW asked to bring his birth certificate. So that is very new to me. I have never had a lap baby I have always bought my little ones a seat.
I always bring along a copy of the birth certificate. My boys are adopted and of another race (The oldest two. I have never traveled with the birth kids) and I got questioned closely one time so I always have a certified copy of their birth certificates. It just makes me feel more secure. Of course now they can talk much better than when this happened and the whining "MOOOOOOOOM he touched me" usually associates them with us pretty well.