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.
YES. **YOU ARE REQUIRED BY INTERNATIONAL LAW TO BRING IT**. You must present passport upon embarking on the vessel as proof of citizenship and as photo ID, which gets entered into the cruise line's computer system. Every time you leave the ship and enter and island you must swipe your ship/room card, which has that same photo ID on it. And since eastern caribbean cruises go to different islands as well, you need that proof of citizenship as you are technically entering a new country.
You will be denied access to board the ship if you do not bring your passport.