Well, I survived New Orleans and Baltimore. I guess I can survive the Caribbean. 
That's true everywhere. In just about everywhere I've worked, there were two ways people got murdered.Stay away from drugs and crime and you cut your risk much more
Travel in the interior can be much safer as well. Often the cities are the most dangerous places. You can't make broad generalizations. We have travelled for many weeks in the interior of countries like Ecuador and South Africa and not seen (or heard of) any reason to be fearful. On the other hand, big cities like Guayaquil, Durban, and Joburg are dangerous places.Homicide and petty theft are on the rise worldwide.
Can't comment on petty theft, but in most places in the world homicide rates are declining (there are obvious exceptions). It comes with populations aging which is happening just about everywhere. It is strange, we seem to want to think that things are getting worse even when the evidence does not support this.
I've been to some of the worst places on earth and there are things to avoid. (i was armed of course; your not)
Stay away from drugs and known drug areas.
Totally agree
Gather Intel on the places you visit. Knowledge is king. Plenty of websites for info.
Agree on the need to get the knowledge, don't agree that the Internet is the best place to get it.
Talk to the locals. They will tell you explicitly where not to go; easy.
If you are cruising there is a very good jungle telegraph of intelligence to be had as well. Local people are terrific and want visitors to their country/town to be safe and enjoy themselves
Don't go out at night; stay away from alleys. Drunk by yourself in a alley is a sure thing to be rolled.
I don't often get drunk anywhere but do go out at night if a place appears OK. Some places it is just unsafe to be out at night but this is the exception rather than the rule.
Travel in groups. Discuss what to do if something happens. Have a plan.
Groups are good, but not always possible. Just be sensible. I have traveled some places with a dummy wallet with a bit of money in it and some expired credit cards and ID (like library cards)
Dress like a local ! Yankees stand out worldwide. Don't look like a tourist! I love to watch americans when i travel. The look like buffoons.
Impossible to do. I don't care how someone dresses they are not going to look like a local in most places. They are often the wrong size and shape, the wrong colour, and have different body language. Try not to stand out too much and avoid the bling is about all you can.
Traveling in the interior of a lot of countries can be MUCH worse. Learn how to travel before you do. Even the local "police" will roll you.
Damn, that crowd of cruisers in the Boot Key KOA Kampground must be awfully deep sleepers...Now, while in Marathon's Boot Key Harbor, there was a series of thefts that took place from the boats of visiting cruisers. A couple locals decided to become industrious and found that generators, dinghies, fishing tackle and outboard motors were easy pickings in the middle of the night. Over a period of two months, they managed to steal more than two dozen generators, a dozen dinghies, 200 fishing rods and reels, several tackle boxes, 30 outboard motors, and more. The perpetrators were eventually caught with the diligence of a dedicated sheriff's deputy who spent many sleepless nights waiting in the dark for the thief to show up.
I've no doubt there are nasty places, and nasty people, out there. Some places are nastier than others, and we should always be alert and conscious of our surroundings. That goes whether you're in a remote anchorage or walking down the streets of New York or Toronto. This doesn't mean we should give into fear of the other (not that I think that is what you are saying Gary).I spent a lot of time on the hook last year, often in remote locations where the only boats I saw were a few commercial fishermen and others just cruising along quietly in skiffs. My worry was that someone would board the boat in the middle of the night to steal something, which is very common in many areas, and I would have to confront them. For this reason, I installed a neat, little infra-red alarm that I purchased from Radio Shack for about $39 many years ago. If the beam is interupted the device sounds a loud wailing electronic siren that can be heard at least 100 yards away. I suspect it would deter any boarder in the middle of the night. The only time it sounded was when a great blue heron decided to enjoy it's supper in my cockpit.