You are going to have a blast!!!!
We actually stay on board at least at one port. The public areas are nearly deserted, so are pools, hot tubs...everything. If it is a port late in the week, and we have been there before, we like to stay aboard and just catch up on down time after several days of fun.
That said... if this is your first cruise and you have kids...forget staying on board. I'll bet the kids would not let you stay on board. Find a tour or two and get out there!
We are actually on the same cruise in July. You will love San Juan and St. Maartin. Shopping in St. Maartin is awesome. The kids will LOVE Labadee. Tons of stuff for them to do. On beach tours in St. Maartin, remember, there are nude beaches. You don't want to stumble onto one with the kids... just know where you are going. For your first cruise it's best to stick with the cruiselines tours until you get your cruiselegs.
Enjoy!!!