Make sure to dress somewhat conservatively. Don't wear flashy jewelry or flash money around. Also carry the minimum amount of valuables as necessary - Looking like you are wealthy or have something that someone wants makes you a target.
Also, use covered luggage tags - These will keep your personal information from being exposed to everyone who comes into contact with your luggage.
Leave a copy of your itinerary and your passport, as well as a photocopy of any credit cards you plan to take with you with someone at home. Even if you're not positive that you'll be taking certain on-shore excursions, jot down the ones you're interested in so that someone at home can have an idea as to what you're doing and where you may be.
Don't discuss your travel plans or other personal information with strangers, no matter how trustworthy they may seem. This includes things as simple as not mentioning the time and airline for your return flight home. Be vague when answering questions, and don't feel obligated to answer personal questions.
While it may be tempting to share the expenses for excursions not organized by the cruiseline, it's much safer to stick with the cruise groups as much as possible.
Pickpockets will use many different methods to steal your purse. They may ask for directions, point to a stain on your shirt, use an accomplice, or create some sort of distraction. Be suspicious of everyone - Even the staff.
Make sure you're aware of the local emergency numbers.
Look purposeful when you're moving about. Even if you're lost, try to appear confident in where you're going.
Keep your door locked at all times, and don't let any visitors in your room. In fact, don't even let them know what room you're staying in. While people can be charming and seem trustworthy, it is much safer to meet up with new cruise friends in a public location.
If you're suspicious of someone in a cab, or an elevator, simply wait for the next one. Better safe than sorry.
A simple self-defense move: If someone grabs your wrist, twist your wrist towards their thumb. Their thumb is the weakest part of their grip. Remember that if someone touches you, you have every right to defend yourself.
If you meet up with a stranger that makes you uncomfortable in any way, let someone in an authority position know. File a complaint if necessary.
I probably sound paranoid, and it probably seems like it will be difficult to relax when you have so many safety issues to think about, but your trip will be great. Just keep your thinking cap on all the time!