Follow fire safety rules
The biggest concern on cruise ships isn't rocks or icebergs, but fire. Rules established by the IMO require smoke detectors, sprinklers and other fire safety measures. But you need to play your part. Smoke only in designated areas. Dispose of cigarettes properly (fires have even been caused by passengers throwing butts over the side of ships). And don't try to make your cabin more romantic by lighting candles.
Do your homework first
Surprisingly, there's no one official source where you can check the safety record of cruise lines or ships, but a quick Google search can point you to news stories that will at least alert you to any disconcerting patterns. For ships calling at U.S. ports, be assured that inspectors from the U.S. Coast Guard come onboard every year for crew training and an evaluation that includes checking the condition of lifeboats.
Register your name
If you're traveling overseas, before leaving home register in the U.S. Department of State's free Smart Traveler Enrollment Program. This will allow officials to assist you in the event of an emergency. Keep the name of your travel agent and travel insurance provider (if any) close at hand, in case you need their assistance — for instance, in arranging a flight home.
Use common sense
Just because you're on vacation doesn't mean you can totally let your guard down. As at home, you'll want to avoid risky behavior such as consuming large amounts of alcohol or letting strangers into your cabin. Be mindful of your surroundings. Lock your door and keep valuables in the safe. If there is an emergency, stay calm. Listen carefully to, and follow, all instructions.
Also of interest: 3 cruise ship rip-offs to avoid.