What would be the significant risks of taking children between the ages of one and seven on a 4 night cruise to Cozumel (an island off the coast of Mexico) if the children have never left the U.S.?
Also, what's a "significant risk" to you?
If you'll spend most of the time aboard the cruise ship then I'd say it pretty much doesn't matter where in the world you're cruising; the risks would be essentially the same -- which is to say, negligible or identical to staying in a hotel or holiday resort.
If you'll spend most of the time in a resort on the island then food reactions are the most likely. I don't know the Caribbeans but that's a common thing for Europeans traveling to Egypt and similar countries. Food reactions are best avoided by avoiding tap water and things that have been washed in tap water (like salads etc.). A risk could also be insect bites which might be more severe than bites from insects you're used to from your home country. Again: playground injuries, theft, child abduction etc. is not what I'd call a "significant risk".
If you'll be on your own on the island I imagine that the chance of encountering a risk is higher, simply because you're more exposed to the local area: foods, tap water, insects, animals (dogs, geckos, snakes, whatever), local people.
In any case, whenever you travel to an unfamiliar country you should always check with your government's travel warnings (political unrest etc.) and health recommendations (vaccines etc.) to make sure you aren't walking into unnecessary risky situations.
Cozumel is lovely, but it can be very hot.
Sunscreen, plenty of drinks, a hat and sandals for the beach are absolutely essential.
It felt safe enough and there were plenty of families there (I stayed mostly at an all-inclusive, going on diving tours). As a tourist destination, the main threats are pick-pockets and being ripped off when buying stuff at a market.
As mentioned above, only drink bottled water, avoid street vendor food if it looks like it's been lying around for too long, and be prepared in any case for the runs and vomiting.