Make sure you travel with a valid passport. Also, bring another valid form of photo ID, and photocopies of both IDs in case of loss or theft. Register with the US embassy or consulate before you travel and check the website before and during your stay for updates on weather, travel and security.
If you stick to the resort, you should be fine, but keep in mind that Jamaica does have a high rate of petty theft. Don’t carry large amounts of cash, avoid wearing expensive jewelry on tours or excursions and keep luggage, handbags or backpacks in view at all times. Make sure you book tours through the hotel concierge or a reputable source.