What is the safest way to get from Cancun airport to Playa del Carmen?

Safe from what? Motor accidents? Bandits? Getting lost?

Your hotel might offer private transportation. Otherwise, if you don't find anything you like at the airport, I would just take the public bus to the Cancun station and then transfer to a bus to PdC, since there are lots of them.

Rent an M1 Abrams. Don't leave home without it. Actually I'm not even kidding all that much. I wouldn't want to travel in Mexico these days without an armored vehicle (and even then I'd be cautious).

Hire a private transport van before you arrive. It's also the quickest.