Best caribbean location for long term stay

Each one is different, and it depends on stay like a hotel room. Get a cruising guide for the Carribbean (I like the Maptech ones), then visit each marina's website that looks interesting, usually the rates are posted. Here is Marina Fort Louis in St Maarten, their rates are reasonable and you can play around with their dockage calculater and see what the different time periods net you. For example a pre-paid year only comes out to $979 Euro's a month plus utilities.

Marina Fort Louis | Dockage Prices and Reservation | Marina St Maarten, Saint Martin

Are you leaving the boat there or as a liveaboard?

The nice thing about the Caribbean is that island hopping is so easy, nowhere is very far away and the weather is mostly predictable. If you just want to cruise around the Virgin Isles, that can be a good introduction to the lifestyle but just a few more miles will get you to St. Maarten. From there you can go down island all the way to Trinidad or Venezuala.

There is not a huge gain to be had hunting for cheap fuel, they all are pretty similar prices on the big islands. Marina prices, on the other hand, vary greatly. So too do living costs.

The cruising websites are mostly aimed at sailing cruisers as there are not as many motoryachts in the area, but they are a good source of information on prices and other subjects such as health services or security. One or two sites are full of people who panic if they go sailing out of sight of a Piggly-Wiggly or the wind messes up their new hair style. Be warned!