"Closed Loop" Cruises: U.S. citizens who board a cruise ship at a port within the United States, travel only within the Western Hemisphere, and return to the same U.S. port on the same ship may present a government issued photo identification, along with proof of citizenship (an original or copy of his or her birth certificate, a Consular report of Birth Abroad, or a Certificate of Naturalization). Please be aware that you may still be required to present a passport to enter the foreign countries your cruise ship is visiting. Check with your cruise line to ensure you have the appropriate documents."
For more detailed information, please visit the state department's web page on the matter:
I always recommend having a passport for the unforeseen event that may result in the event you miss your embarkation time or a medical emergency arises which requires a flight. The bottom line is that it is your responsibility to make the decision whether or not to have a passport when sailing on a "Closed Loop" cruise from a US port.
My mantra: Better to be safe than sorry. Always prepare for the unexpected.