Options for transportation:
At the airport there are plentiful signs for taxis to the city which charge $40-$45 USD. There is no bus or train service to the city direct from the airport. We did find however posts online of travelers who had used a shared van service which was remarkably cheaper, around $3-4 USD per person.
Finding the affordable van transfer:
We decided to find and use this service upon our arrival at the Airport of the Americas in Santo Domingo. We arrived around noon and exited the arrivals hall. We immediately proceeded upstairs to the departures floor and asked some airport staff where we might find the van. As we had previously read, they told us to go outside to the very first departures door (as you walk out of the building turn left and walk all the way down).
Using the van transportation to Santo Domingo:
There at curbside in front of the first departures door we found a plain white van which already had some passengers inside. It is a no frills operation, just hop in and share a seat where you can. The driver quoted us the price of 70 dominican pesos per person. If taking a bag it is 70 pesos per bag as well. We were able to pay the equivalent in US dollars. For this process to go smoothly we recommend a working knowledge of Spanish.
Drop off and Airport return:
Once the van fills up the driver takes off for the city. The end drop off point is at a central location in front of Hospital Dario Contrera not far from the Columbus Lighthouse Monument. Some passengers will depart the vehicle sooner if they have other destinations along the way. For a tourist to the city, most likely the last stop will be closer to your hotel. The same bus stop in front of the hospital is where one can take the vans back to the airport. The transportation operates during daytime hours and vans take off for the airport when they fill up.