What cruise line are you going on? The Carnival site have the following information:
[I]U.S. citizens traveling on cruises that begin and end in the same U.S. port must show proof of citizenship (such as an original or certified copy of a birth certificate) and a government-issued photo ID (such as a driver’s license). Effective June 1, 2009, U.S. citizens traveling on cruises that begin and end in different U.S. ports, or begin or end in a foreign port, must have a passport or other recognized document. For a complete list of accepted documents, see Welcome to Travel.State.Gov.
U.S. Alien Residents need a valid Alien Resident Card. Canadian citizens must present a valid passport. Non-U.S. citizens need a valid passport and a valid, unexpired U.S. Multiple Re-entry Visa, if applicable.
Carnival recommends that all guests travel with a passport valid for at least 6 months beyond the completion of travel. For more information or to obtain a passport application, visit Welcome to Travel.State.Gov. [/I]
It sounds like you’ll be fine unless your cruise ends after June 1st. However, a passport might make it easier.