An Alaskan cruise usually only visits the US and Canada. A Caribbean cruise often visits countries other than the US (and, sometimes, Canada). Almost all Eastern Caribbean cruises visit at least one island in The Bahamas. Western Caribbean cruises usually visit Mexico.
You must have appropriate travel documznts (passport, visas when necessary, etc.) for EACH country on your itinerary and your documents must meet whatever requirements exist for that country. If you don't have the paperwork, boarding may be denied, without a a refund from the cruise line.
Safe travels!
steve