Just click on the Weather link above (under Travel Essentials) for average monthly rainfall for the USVI. We have gone during the 2nd week of December for the past several years with generally wonderful weather. And, I would assume the later in the month you go (i.e. the further from November which does average higher rainfall), the better it might get. Enjoy!
Mosquitos will usually follow a rainy period. Best to always use insect repellent. We always use Deep Woods Off which contains a decent amount of Deet (I believe ~25%). Noseeums are usually out at dawn and dusk so best to not be on the beach during those times (or if so, definitely with repellent).