Make sure that the birth certificate is one that is valid for identification purposes (raised seal), also YES you will need a photo ID. I believe that the Motor Vehicles Department in many states can issue one for someone who does not have a drivers license or can tell you where to get one.
A photo ID is necessary and must be issued but a form of government such as Dept of Motor Vehicles. Take her and her birth certificate to the DL place and get the state ID. You will need both. After the cruise go ahead and get passports for everyone, you will never regret it>
The state issued ID's usually only cost about $5 and don't take much time at all to get. I have an aunt who is legally blind, so she doesn't drive, and the state issued ID worked great for her on the cruise. In some states the ID# is what your drivers license number will be if/when you get one.