I've heard of 'sand flea' problems in central america .. either I don't know what they are, or LC / Brac doesn't have a (significant) problem with them. My "bug-sweet" wife tends to sit on the beach all morning while I'm diving and while she occasionally gets a couple of bits on the shins, its hard to say what they were.
You will certainly find mosquitos under the usual conditions ... sunset, particularly when there's low wind conditions ... as well as probably some other critters. I've never seen biting flies in Cayman (such as you find in many areas along the US Eastern Seaboard). As of a month ago, they were having a pretty severe mosquito outbreak.
One of these critters that one should keep an eye out for was locally known as a "coconut bug". To me, it looked to me like a lightning bug (same look & shape, but slightly smaller). Its noteriety is because its juices are acidic and can cause a mild burn if you squish/kill it on you - - so the trick is to not kill it (at least while its on you), but to sweep them off. Oddly, they have been rampant on one of my LC trips and absent from others, so I don't know if they are seasonal, or that I simply hit one single year with an unusually high spike.
In general, I'd recommend taking some bug spray, be aware that sunset & dusk is the hour that they'll be most active, and to look before you smack: if its not a mosquito, brush it away.
After a couple of days & sunsets, you'll have a good idea what the rest of your week is going to be like and to plan accordingly.
-hh
