I don't think Jamaica is anymore unsafe than most international travel destinations. Yes, it is poor, yes there is security, but in most places (especially near airports/customs) armed guards are not uncommon. As with any travel, you should use common sense (don't travel alone, don't go far off the beaten path without knowledge of what's out there, etc.).
We honeymooned in Ocho Rios and it was delightful. We did some off-site tours through the hotel, which were fine. Tourism is a big industry in Jamaica, and a huge boost to the economy. I would suggest a nice resort that you will be happy at, we didn't leave the resort too much, but that is because there was plenty to do.
Note: having a decent amount of international travel experience, I am probably more comfortable with cultural differences than many. DH, who hasn't traveled as much internationally was less at ease with some of the customs (e.g. more aggresive sales tactics at the market) and prefered to stay at the resort, but he never said he felt unsafe or anything like that.