There's no train connection with Santo Domingo Airport.
Car and bus
Las Américas Airport is situated 20 kilometers southeast of Santo Domingo. The capital is easily accessible by rental car. From the airport a freeway runs across the Ozama River to the main street of Santo Domingo, Avenida 27 de Febrero. Although this is a quick route, the road along the coast is much nicer. This one leads to the Avenida George Washington and on to the center of the old city (Ciudad Colonial).
There's also an (irregular) bus connection between the airport and Santo Domingo's city center. Minibuses also run to and from the airport.
Car rental Santo Domingo Airport
You can find counters of car rental companies in the Arrivals Hall. Many rental companies are present at Santo Domingo Airport, including local providers and international ones such as Avis, Budget, Europcar, National and Sixt. You can also book a rental car in advance using the service provided by EasyTerra; they compare the rates of several providers in a single quote.
Taxis can be found outside the Arrivals Hall. The costs of a ride to Santo Domingo's center mainly depend on your negotiation skills. If you can manage to knock down the price you can make the taxi ride for 500 pesos; if not, you may pay twice that much. A taxi to Santana Beach costs about 1,500 pesos. Taxi drivers also accept US$.
Opposite the terminal lies a parking lot which can accommodate about 1,500 cars. Part of it has been reserved for rental cars. You pay 20 pesos for the first hour and, after that, 15 pesos per hour. A full day parking costs 160 pesos for the first day, and after that 60 pesos per day.