Actually the SAFEST places are off the beaten path. What's called the "bush". These small town's rarely see tourists and aren't expecting you when you show up, so they are both curious & very nice...of course their accent is alot thicker so understanding may be a problem, but "country people " are warm and friendly and you'll never have the begging that happens in the tourist areas like Ochi, Mobay, Negril. Though driving the interior of the island has it's risks, mainly no road signs an only an occasional direction sign, but being lost is part of the fun & safe, as asking someone the way home works well.
As mentioned, Port Antonio is nice, as is Black River, Mandeville, Savlamar, Falmouth and a few other town's that attract tourists, but they are increasing their tourist traffic which increases the bad elements too. Understand that the money attracts outsiders so Negril, Montego Bay, Ocho Rios has people not from there that come in trying to make a dollar and bring in their own form of "crime" which could be as simple as stealing your stuff while you're at the beach & not paying attention.
As to Sting's comment about "natives", that's ridiculous...people call themselves natives all over the world... " I'm a native Californian " or if someone moves from New York to Florida " I live in Florida, but I'm a native New Yorker " is said all the time. A native only designates where you're from...period.