The nearest airport is Rafael Hernández Airport in Aguadilla (IATA: BQN), about 30 minutes north of Rincón. Several airlines offer direct flights from New York, Orlando, and other US cities. San Juan's Luis Muñoz Marín International Airport (SJU) is larger with more connections, about 2.5 hours east by car — a reasonable choice if you are combining Rincón with time in San Juan.
From the airports, car rental is the practical choice. There is no public transport to Rincón. The town is spread out and a car is necessary once you are here to reach the different beaches and the lighthouse. Driving from San Juan takes around 2.5 hours via PR-22 and PR-2 — the road is good, the highway tolls are paid by transponder.
Rincón has no bus or taxi services of note; rideshares (Uber, local apps) function but coverage is limited. A rental car for the duration of your stay is strongly recommended.