Your best bet is to check with where you are staying and ask them who they recommend. Most guest houses, hotels and villas have drivers they recommend for airport transfers and tours. Some are Jamaica Tourist Board licensed, some are not. Another option is negotiating with public transportation drivers to charter (hire exclusively) them for your trip. See also Public Transportation.
FYI: Only vehicles with red license plates are properly insured to carry public passengers. White license plates denote a private vehicle which may not be properly insured.
Treasure Tours is the only Jamaica Tourist Board approved tour company in the area and can be hired for airport transfers and day trips. They also book select accommodations.