I went to new Orleans and brought my 7 months old babies birth certificate but I ended up not needing it
I flew from Oklahoma to California 2x with my son under the age of 1, once at 6 months and once at 11 months- and needed the birth certificate to present at check in. They asked every time.
I take them just in case. About half the time I have to show TSA. If you don't have it, some TSA won't let you through. So it safer just to have them with you.
Just take a picture of important things