While there can always be bad guys anywhere you go, we've been going to Cozumel since 1997 and always felt safe. I've ridden a bicycle around the island by myself. We've done extensive walking far beyond the tourist area (you may get curious looks, but a simple "Hola" is usually met with a greeting and a smile). My wife goes bargain hunting out on her own and has always felt comfortable.
Honestly, the most apprehension that I've had is: (1) dogs - common to see unleashed dogs, and 99% won't be a problem, but I did have one chase me on a bicycle on a residential road near El Cedral; and (2) again on a bicycle, if you go way, way up the dirt road to where the Passion Island boat docks I've seen a pack of raccoons there rummaging for garbage that made me a bit nervous - they were not frightened by humans. Both of these are out of the way from where most tourists would be.
Beyond that, be smart about using ATMs at night, watch for scooters passing on both sides of your vehicle, get the taxi driver to tell you the cost before you get in the cab, avoid timeshare salespeople (at the airport, and everywhere else), and watch for Topes (speedbumps that are everywhere and range from almost flat to big plateaus; many are marked with signs, but many are not). Overall, relax and enjoy -- it's a great place!