Here is the link to the current State Department information sheet on Belize (found with a simple Google search).
It says that U.S. citizens must have a passport to enter. It also gives the link to the Embassy of Belize for further information.
A cruise line or a travel agent is the last place I'd look for valid information about entry requirements. The most reliable source is the embassy of the country in question; the next-most-reliable is our State Department, which makes this information readily available.
I'd call or e-mail the embassy.