Yes, I know you can get food poisoning anywhere, even your own backyard.
But generally speaking, what island(s) do you think have cleaner water, better hygeine for food prep, etc.
Interesting re-reading the OP because it really is two separate issues - food
and non-food sources of potential gastrointestinal illness/infection.
Both can happen anywhere. Norovirus outbreaks at WDW and on cruise ships are good examples of how GI illness can spread like wildfire even if you don't eat something that makes you sick. (Though originally, those germs came from a GI source somewhere.)
There are some things you can do to protect yourself and your family no matter where you are.
Practice good hand hygiene and keep your hands away from your face, nose and mouth. Teach your children to do this and call their attention to it if you see them doing it inadvertently.
Be careful what you touch in a bathroom (any bathroom, but especially public ones). The last thing you do in there should be using a paper towel to exit with freshly washed hands protected. Things like latches in stalls, knobs on sinks, and exit door knobs are saturated with germs as people touch these after wiping and some don't wash afterwards, and whether they're ever or thoroughly cleaned is sketchy at best. Aerosolization occurs when toilets flush so there germs are on most surfaces in a bathroom. (And recent studies on Norovirus type infections have confirmed aerosolization as a source of transmission when previously it was thought to be mainly via hand to mouth.)
Do not eat undercooked meat. And be aware of
where you eat - food should ideally be washed and prepared properly, so stick to the better food establishments. (Though food-borne illness can occur even when proper washing and food prep happens.

However your chances of getting sick statistically are probably less at places that follow proper guidelines.)
Fruit and produce are as dangerous as meat as generally there is no "kill step" where it's heated to kill bacteria that may have contaminated it (either from being washed with contaminated water or cut up on a contaminated surface, etc.). Be careful with these. (Momtobrent had the right idea here.) Lots and lots of infections here in the US with things like sprouts, cantelope, bagged salad, etc.
I haven't traveled much internationally so I can't speak to the water issue.