The town beaches in both Capurganá and Sapzurro are lovely and relatively clean. Sapzurro's is much quieter and complemented by the wilder Cabo Tiburón about 15 minutes' walk around a rocky peninsula.
A short hike over the hillside from Sapzurro into Panama takes you to the most famous beach in the area, La Miel (bring your ID – there's a military checkpoint). It's a quick walk up a series of steep steps across the border and back down the other side (turn right at the bottom of the hill and follow the sidewalk). The small beach has white sand, cerulean waters and several small shacks where you can eat fresh fish and drink a cold beer. Sadly, it's a little marred by litter and noise pollution at times.
Walk for 30 minutes along the coast beyond Capurganá's town beach and you'll reach the beautiful rocky lookout of Coquitos, where an enterprising local man has built a rustic set of swimming holes (two freshwater and one saltwater). He charges COP$3000 entry and serves fabulous snacks with homemade lemonade. It's a fabulously tranquil spot.