It depends on the cruise itinerary. If it is a closed loop cruise then the baby's original birth certificate will suffice. If it is not a closed loop cruise, then regardless of age a passport will be needed.
Look here for the definition of a closed loop cruise.
You should have a passport for the baby no manner what. If you have to return home from one of the ports you will be out of luck as the baby will not be allow to get on a plane. Passports are the safety and secure way to travel. My family have always gotten passports even when everyone was a baby.