Yes. Depending on which port you are docked in you can walk or take a short cab ride into Old Town. (The old port is closest.) Streets are fairly safe, but you do get your panhandlers who will approach palm out, with some change in their palms, asking for the balance for bus fare. Handle them as you would handle panhandlers anywhere. Act as you would in any city and you should be OK. You'll find charming blue cobblestone streets, little shops, bars, art galleries, jewelry stores. Plus if you forgot something like toothpaste or aspirin, there's a Walgreens with all that stuff at reasonable prices. There's a square with a fountain and street vendors and at certain times of the year, there is music and fireworks.
If you have more time you can visit the Baccardi rum factory--interesting and free samples! If you have even more time you can visit the Condado district--hotels and casinos.
If you are in early in the day and want to go to a nice spot for lunch try El Convento (a converted convent),
And of course, there is El Moro, a big fort that dominates the coastline. It's huge and can take a bit of time to explore. Maybe you can save this for when you return to port.
Sailboat - You sound pretty familiar with San Juan. Do you know much about the hotels? We are flying in 2 days early and have not decided where to stay yet. We don't need any of the fancy hotels with casinos, just a clean place in a safe area. Any suggestions?