We've been to Puerto Rico many times. Like many places around the globe, vigilance is important and basic safety measures are vital. These include: not appearing too "tourist-y", like wearing all American baseball team clothing or t shirts with suggestive or offensive language. Wear basic clothing - plain t shirts or blouses, ordinary jeans, good walking shoes, nothing flashy or sloppy. Don't purchase tons of souvenirs that you'll have to carry around - that identifies you as a tourist.
Keep your IDs in a secure spot - under your clothing. Don't wear fanny packs or put your IDs in a wallet in your pants pocket. Purchase those ID holders that you wear around your neck under a shirt, or as a belt. REI and Magellan are great sites for secure travel pouches and purses that can't be pickpocketed.
One thing we noticed on the highways around San Juan is that many of the exits aren't named or numbered. That makes things interesting! We resorted to counting exits. But driving is ok as long as you're not expecting clearly marked exits. Trial and error!
If you want to truly experience the island, find a place to stay near old San Juan and walk and explore. It's beautiful. El Moro (the old fort) is gorgeous. The rain forest, El Yunque, is the only rain forest in North America and it's a magnificent experience. Our kids were young the first time we went there and we have wonderful photos of our son holding our daughter's hand as they played in the natural pool at the rain forest.
Don't stay in an inclusive resort, if you want the real experience. Those are very structured and intended to create an ambience. We loved staying in a local hotel, eating street food, buying a souvenir from a local woman who made hand-woven purses and was selling them from a pallet on the street, and drinking coco frios (ice cold coconuts, with the tops macheted off with a great display). One of our favorite photos is of our daughter, just 4 at the time, sipping on a coco frio, making the worst scrunched-up face. We paid $20 to a street vendor so the kids could pose with beautiful friendly parrots on their shoulders - yes, a tourist-y thing, but nothing we could ever replicate.
Just stay smart, stay alert, don't be afraid. Respect the locals. Go to Barrachina, the restaurant where (allegedly) the Pina Colada was invented. Our kids loved seeing the turtles in the ponds, the birds, and they loved the virgin pina coladas there.
There's crime everywhere, from Nebraska to Maine to California. Don't let that stop you. Be safe. Be smart. Don't look like a tourist. Blend in (plain clothes, not a lot of shopping bags). Keep IDs under your clothes and make sure a family member back home has copies of your itineraries, tickets and IDs. Don't fall for pickpocket schemes (people trying to sell you flowers who wave the flowers in front of your face, people who create a distraction such as dropping a briefcase full of papers and appearing very upset, people who scream that they've been robbed thus causing everyone to double check their wallets and alerting thieves to the location of easy targets).
Puerto Rico is beautiful.