I have yet to hear of a crime aboard any Princess vessel. I won't say that it can't happen, but you don't have to worry about it.
>> 1. Princess attracts a mostly upper middle class cadre of passengers who generally are not the sort of people who would commit crimes. The line also does not have a reputation as a "pick-up" spot for singles (for whom most pundits recommend Carnival), so it does not many "lounge lizzards" and other varmin from Animal House.
>> 2. Princess screens their staff very thoroughly to ensure that they are trustworthy and maintains a very strict "zero tolerance" policiy with regard to any sort of misconduct by the crew and the staff. Any member of the crew caught in a passenger stateroom outside of duty would be left ashore at the next port, whether the passenger was there at the time or not -- and the members of the crew know it!
>> 3. Princess also has a very competent and adept security team aboard every ship, augmented by technology that allows them to monitor the entire ship and to respond immediately if there is a problem.
The result is that Princess provides about the safest environment that you will find anywhere.
If you are anxious about safety in any ports of call, just take a tour from the ship. There's tremendous safety in numbers and they won't take you to any areas that are dangerous. Also, Princess will cancel any port of call where there's unrest or civil strife, replacing it with either an alternate port of call, if possible, or a day at sea. You'll be safe.