We made port in Martinique
at Fort-de-France Bay, near Fort St. Louis, one year while on Royal Clipper... and maybe that's where other cruise ships enter...
It's not far from the airport, probably an easy enough taxi ride. Google
Maps, zoom in enough, and you can see where the Cruise Terminal is located... and the relationship to where the airport lives.
Maybe the tourist bureau there would be a source for better info for ground transportation... and the ship's purser may even be able to arrange in advance a taxi waiting and ready....