June through November will be good for most places, but as @KathyV said, it is hurricane season. Bonaire and Curacao are not as susceptible to hurricanes as most of the other islands. I was in Turks & Caicos last November, and the water was great. It was a little disappointing that some of the places we liked were still closed due to hurricane damage, but I am sure that most things are open now. Kathy is right about Grace Bay beach. My friends have been there February and they said the water was too cold.
We were in St Lucia last month, and the water was spectacular. I only did two dives in St. Lucia, but Superman's Flight was one of the best dives I have done. Easy drift dive and the corals, sponges and fish life was just awesome. I thought it was much better than Grand Cayman (did ten dives there). The plus for Grand Cayman was Stingray City. I have done both the dive and the sand bar and both are worth your time. While I thought the corals in GCM were great, it just seemed devoid of reef fish.
Funny, but I would have said the exact opposite. We thought that St. Lucia was beautiful and the corals were lovely but they seemed overfished - lots of small animals but not many large fish. We described it as "a good environment for macrophotography."
The Caymans always seem quite fishy to us with lots of bigger animals like groupers, eels, rays, tarpons, turtles, barracudas, and some sharks - although it is not a fishy as Florida IMO.
Maybe it has something to do with season and water temp? We were in St. Lucia in the Spring and we usually go to Cayman in the Fall.