The only passengers required by TSA to present ID are adults 18 and over. We have never been asked to present any ID for our kids, except when we went out of the country. Never on domestic flights. Security usually asks them their name but that's it. It doesn't matter which airline you fly on...the security checkpoints are run by TSA not the carriers, which is probably why Southwest wasn't very helpful.
http://www.tsa.gov/travelers/airtrav...documents.shtm
Here is a page from the TSA that covers "traveling with children." HTH
http://www.tsa.gov/travelers/airtrav...ren/index.shtm