It apparently depends on your nationality in some places. For U.S. citizens in The Bahamas, a Photo ID for those under 16 is not required to re-board the ship. All you need for the children is their cruise cards. For the adults, a photo ID (license) will do. You do not need to have U.S. passports with you while ashore.
They don't look for ID when you exit the ship, just when you return.