The concierge at a lot of high end resorts will refer you to a babysitting or nursing agency if they do not handle this in house.
Ritz Calrton and Four Seasons hotels tend to refer to nanny services in the local area. I have used them in various beach, city, and ski locations, and been perfectly happy with them. The rates vary - you're usually looking at a three or four hour minimum, with the charges being between $15 and $20 per hour. We've used their recommendations in Sarasota, FL, Miami Beach, FL, and Philadelphia, and explored the options and availability in Bachelor Gulch, CO.
Mexican law prohibits children who are in diapers from using daycare facilities. The Rosewood and the Banyan Tree, both located in the Mayakoba resort in Playa Del Carmen, provide sitters from an in house rotation. We've used the sitters in both locations and been very happy with them. $15-20/hour, three hour minimum. They can also be flexible if you like, especially if they handle the sitting in house.
In most of these instances, we've used the sitters in the evenings when we want to head out for dinner. We'll put our child down to bed, then have them ensure that nothing happens when we're out.
This winter we decided to bring a babysitter from home with us when skiing, as our son was comfortable with spending time with her during the day, and it worked out for everyone from a financial standpoint.
Certain resorts in Jamaica and the Caribbean will also provide nanny's upon request, sometimes for an additional fee.