I don't think the water is dangerous to healthy people, but if you have a desease that weakens the immune system - take precautions, always, not only in the Caribbean... I am not a doctor, but as a reporter I have interviewed several experts on the subject of dangerous tap water. This is what I learned:
Quantity... the amount of water that you would swallow brushing your teeth or showering is probably not enough to make you sick even if there were a pathologic type of bacteria in it... drinking the water cold/lukewarm by the litre/gallon would be another thing...
Resistance... if you are healthy otherwise, you may not be sick at all... Acidophillus pills is an easy preventative
Note... that all desease carried in water is not infecting through ingesting, but may infect through inhaling, such as legionella that spreads in water mist...
Legionella as a bacteria is very common, but the desease is rare. You would most probably not get it at all if you are healthy... being of poor physical status and then being exposed to the disease is dangerous though... in that case you may be cautious about showering and checking that the water is hot enough - even at home. It's easy to prevent from getting potentially dangerously high levels of legionella, make sure the hot water is hot enough, and run a little hot water through the shower tap before mixing hot & cold for showering, and have a good routine for cleaning hot tubs etc.
As far as I know, there is no dangerous tropic desease on St Barth, such as Bilharzia, in the waters.
That said, since I am reasonbly healthy I will not worry about showering, jacuzzing and brushing my teeth in tap water or even boiling my egg or pasta in it, but will choose bottled water for drinking cold, as it probably tastes a lot better
/K