If the answer above is None it means there is no specific mention in the legislation, but you still need to ask local law enforcement before drinking in Cayman Islands.
It is prohibited to sell, serve or supply alcohol beverages to anyone under the age of 18 years. It is also prohibited for any minor to buy, attempt to buy or consume alcohol beverages. Violation of the law is punished with a fine of five thousand dollars, and the court may, if the offence was committed by the licensee, order that the licensee shall forfeit his licence and that no licence may be issued to that person for a period not exceeding ten years from the date of his conviction.
You can find more legal details and links here:
Wikipedia Legal Drinking Page