I've been to St Kitts and Barbados several times each on cruise ship stops, diving every time and never had a problem.
In St Kitts I've used Pro Divers and Dive St Kitts. The Pro Divers shop is located on the west side of the harbor, while Dive St Kitts is east of the harbor (where the cruise ships dock in Basseterre.) Both shops provide transportation between the ship and the shop, and each is around a 5 minute drive.
Of the two shops, Pro Divers is more formal and Dive St Kitts is more laid back. I had good experiences with both... but personally I would go back to Dive St Kitts because I don't need to be baby-sat (and Pro Divers seemed to assume that they had to babysit everyone.) As for the diving... there are a handful of shallow wrecks nearby (including a bulldozer that somehow fell off a barge many years ago), and plenty of reef sharks too.
In Barbados I've used Roger's Scuba Shack the past couple times. They picked us up from the cruise ship terminal and brought us back; it was a relatively short drive (around 20 minutes.) Roger and his brother have been running their scuba business for nearly 40 years. It's a very cool shop with a great vibe. The guys crewing the dive boat were hilarious, in a good way; they knew how to make sure everyone had a good time while also taking care of their business (i.e. safe diving.)
As for the diving in Barbados... it was okay. There were a few decent wrecks, a fair amount of life... not the ultimate Caribbean diving locale, to be sure, but not disappointing either.
I've been to Antigua only once, many years ago, on a cruise ship stop, and it didn't make enough of an impression for me to offer any useful opinion. Never been to Grenada.
Hope this helps! Whatever you decide for your diving, I'm sure you'll enjoy.