Yes, Playa del Carmen is very safe for travelers and local residents. As well, Cozumel and all of the resort areas along the coast of the Yucatan Peninsula are considered to be very safe travel destinations. Of course, you should observe the same safety precautions here as you would anywhere else. Taxi cab drivers are very friendly and quickly reassure visitors that they're very safe in this part of Mexico.
You can not avoid the fact that there is a major problem in Mexico with drug cartels, street crime, and violence in Mexico. The U.S. Department of State has issued warnings advising travelers of the dangers in certain parts of Mexico. Most of the violence is concentrated near the border between the United States and Mexico and along several well-known drug corridors. Those cities are well over 1,000 miles from Cancun, Playa del Carmen, and the Riviera Maya.
There is no need to be overly concerned that you might be a victim of crime. The Yucatan Peninsula is a safe and relaxing place to unwind, enjoy the great beaches, perfect weather and forget about all of your troubles. Crime is not a major issue in the tourist areas of Mexico.