If you are leaving from a US port and returning to the same port at the end of the cruise (which most cruises do), you can do it with a certified copy of your birth certificate and a government-issued photo ID, such as a driver's license.
This is true regardless of which cruise line you go on, and regardless of whether the cruise goes to the Eastern Caribbean, Western Caribbean, Southern Caribbean, the Bahamas, or Bermuda.
HOWEVER, if something goes wrong on the cruise (you miss the ship, you need to go home early due to a family emergency, etc), you will encounter difficulty flying home from one of the ports, because a passport is needed for all international air travel.