I am asked this question all the time. It is important to know that it requires the father to be participating. A mother can list him on the application but he must be involved for it to make it to the actual birth certificate.
A benefit for including the father on the birth certificate is Social Security benefits. Your child would be eligible for Social Security death benefits if needed later.
Listing him on the birth certificate does not allow him to get custody. He can pursue custody or visitation regardless if he is listed on the birth certificate. He would do this legally.
How DO I ADD HIM TO THE BIRTH CERTIFICATE?
You would both sign an affidavit legally acknowledging him as the birth father. The hospital can provide you with the proper paperwork after the birth.