Also never have I come across any airline staff requesting ID from any small child for a domestic flight. It is NOT a requirement unless it is an international flight.
Anyway why say anything if not asked?
Most airlines require you to tell them if you have a lap child when you check in for your flight. SW gives you a special boarding pass for the child. In order to get that pass, they ask for proof of age.
From SW:
If you choose to travel with your infant on your lap (at no additional charge) the infant will not need a boarding pass; however, you will need to obtain a Boarding Verification Document (BVD) for the infant prior to proceeding to the security checkpoint. The BVD will allow the infant to proceed through security screening and to board the aircraft. BVDs are available at the Southwest Airlines Ticket Counter on the day of travel. In order to complete your BVD, the Customer Service Agent will need to verify that your infant has not reached his or her second birthday, so be sure to bring along a copy of your baby’s birth certificate.
From US Air:
Infants
US Airways defines an infant as a child less than 2 years (24 months) of age. Infants less than 2 years of age must be accompanied on all flights and in the same compartment with an adult at least 18 years of age. US Airways recommends traveling with a birth certificate for a lap child between the ages of 1 and 2.
From Jetblue:
Customers traveling with lap children are required to call 1-800-JETBLUE (538-2583)
to provide JetBlue with the name and birthdate of the lap child. Customers traveling with a lap child could be asked for proof of age, such as a passport, birth certificate (copies are acceptable for domestic travel) or an
immunization record when the infant's age is questionable.
Airtran:
One child over 14 days and under two (2) years of age, not occupying a seat, may be held in the lap, free of charge with a customer paying the applicable adult fare. A birth certificate may be required to validate the age of all children under the age of two.
I know on our trip in Feb of 2007, when we went through security, I didnt grab DS2's BP out of the bag by accident. The guy checking IDs and BPs asked my husband where it was and I had to dig it out. We have always bought the kids tickets, so I didn't know if they had to have some sort of boarding document for a lap baby or not.