Matt,
To be frank, insects are an issue in many tropical destinations, not only in the Caribbean, but other tropical destinations as well. It's part and parcel in many cases. Some places do a better job at spraying but it has to be done carefully so as not to damage the reefs and the surrounding ecosystems. Some locations have insects but they also have more winds which keep the bugs away.
Roatan is a great diving destination but has a big problem with no-see-ums. As a rule, I do not lie directly on the sand when I am there. Puerto Rico has mosquitos but they are mostly a problem in the evening, not so much during the day. Aruba, Bonaire, and Curacao have bugs but they have strong tradewinds. Grand Cayman has bugs but they tend to do a better job at spraying from what I've heard. If you search the boards, you'll see that St. Maarten/St. Martin has a mosquito problem, but it's one of the top gastronomic locations in the Caribbean so it gets a lot of tourists.
When I go to a place with lots of bugs, I buy repellent and wear clothes that offer some protection. Also, as far as the mesh over beds, that's also needed because some locations do not necessarily offer air conditioning so the windows are kept open for a breeze. If your lodging includes ceiling fans, that will help as well. I do not like insect bites but I think you can minimize the impact by taking a few precautions from the travel web sites. On the other hand, if you have severe allergies, that's a whole different matter.
Hope that helps!