Hotels in San Juan are expensive. I have never stayed at one but will be on the Destiny in June and was doing some research re: hotels in SJ. I found the cheapest hotel was approximately $150 a night. However, you get what you pay for. Staying near the ship's terminal will also cost you, especially the night before the ship pulls out. Try tripadvisor.com. They have loads of customer reviews and you should be able to find something that fits your budget.
The only island I have strong recommendations on is Barbados. When I went on the Destiny in 2006, I went to Boatyard Beach in Barbados. It was and still is the most beautiful beach I have ever seen. I did not know that sea water could be that clear and calm. It was like standing in a giant bathtub. The sand is truly of the texture of talc power. There are loads of free water toys to play with (trampoline, swing rope, floating climbing triangle, kayaks, etc.). There is a bar/restaurant that sells very reasonable and really good food and music playing by the bar/restaurant. There are also several other restaurants up and down the beach but they are decidedly more expensive (lobster was around $40 at one restaurant). There will be people walking up and down the beach offering to take you out to snorkel. I don't snorkel but after a while I hired a glass bottom boat to take me out to see the "sunken ships". It was pretty cool. The young man driving the boat took me out pretty far and then gave me some bread to hold in my fist in the water. A whole bunch fish came and ate the bread out of my hand. This was pretty cool for a gal hailing from Brooklyn, NY. The admission for this day in paradise was around $15-20 which included a ride back to the ship (glass bottom boat was another $15).
St. Thomas has Megan's Bay which is also lovely not as pretty as Boatyard but wonderful none the less. Make sure you make time to go downtown to wonder up and down streets and alleys to look in the shops. It was wonderful and I never felt threatened or unsafe.
Loved Dominica and plan to do the river tubing excursion (didn't do it last time). This is such a wonderful unspoiled island. Many people said the people were poor on this island but I think poor is a relative term. I think their lives was more simple than poor. I didn't see poverty as much as I saw blessed simplicity.
After my tour, I wandered alone through the downtonw area near the ship's terminal and never felt unsafe and everyone was very nice and pleasant.
On Antigua went to Turner's Point Beach. It is a very simple beach with nothing but a small restaurant and a few thatch shelters for your lounge chairs ($5). This must be one of the beaches that was used for those Corona commercials because this is exactly what it looks like -- a scene from a Corona commercial. Very simple but very very nice. Next time around I will try another beach but I can honestly recommend this one. The food at the restaurant was very good and cheap.
Can't remember the other islands you will visit but I have never been to St. Kitt, St. Croix or the DR.