A Key to the World card is always required for each passenger to reboard the ship.
At most ports, the only other requirement is that those 16 or older have a government-issued photo ID, such as a driver's license or passport.
A few ports specifically require a passport for all passengers to return to the ship. The day's Personal Navigator and the Cruise Director's speech at the shows the previous night will let you know if Bermuda is one of them.