I'm looking into departing on a cruise from Florida that goes through several Carribean nations. Given that I'm not a US citizen, there are a couple of stops on the itinerary where my passport alone won't suffice. Would the cruise company allow me on board despite that? And would I be allowed to exit the ship at ports where I would otherwise require a visa?
I've been on several cruises on two different cruise lines and the rule has always been as follows: you are responsible for getting your own visas (and for knowing which ones you require); the cruise ship staff will check your passport at check-in time (on embarkation day) to make sure you are allowed in all the countries where the ship will stop. You will not be allowed on board unless you meet all the requirements for all the countries, even if you don't plan to leave the ship. If you try to book a cruise through a cruise line's website, you will probably be given this information before you complete your booking (in case you want to verify my answer).
Norwegian's visa page has some helpful resources. In general, if you require a visa to enter a country, the cruise line is usually going to require you to hold that visa to board the ship at all; it's not usually an option to say you just won't get off the ship.
Note that some Caribbean countries (e.g. Bermuda) have visa exemptions for those holding multiple-entry visas from countries like the US and Canada, so you may not need a visa even if your nationality would otherwise make one required.
When I went to the Carribean a long time ago (with a UK passport) we had to get a Visa from the country we started the cruise in. This Visa head to last the length of the cruise, but was easy being British as we could get it on arrival at the airport.
Each Island the cruise visited, the captain did all the immigration paperwork for everyone. The captain kept all of our passports and issues as with a ships ID card that allowed as to enter the Island. Unlike getting a normal visa for each island, there was no direct cost to us of this process.
When we left the ship we were ticked off on a list, and they would only allow people to leave the ship who were approved to enter the given Island. (Some of the staff were not.)
However being British we qualified for automatic visas for all of the islands anyway without need pre approved, so I don't know how it would have work for a person with a "less powerful" passport.
I suspect that the answer will depend on your tour operator and travel agent. In the case of a UK based cruise, it is possible that you need only arrange a passport; the booking agent or travel operator will do the rest. You will need to check with your travel agent or the cruise operator for confirmation.
When I went on a Caribbean cruise with a well-known UK cruise operator they arranged all the visas as part of the booking. I am a UK citizen with a UK passport and I had to make sure I had at least 6 months on the passport left at the time of my 2-week cruise.
At many of the docking points, there were no port staff actively checking passports and visas. The crew of the boat worked with the local authorities to check you on and off the boat in an acceptable manner. The boat operator had either got a waiver or had applied for visas on my behalf. I had to carry my passport on some islands and boat ID card on all.
My cruise started and ended in Barbados. I flew into there from the UK, went through a long ID check and security process once, then boarded the boat. This process was repeated again at the end of the cruise.