If either parent had US citizenship when the child was born, their children may automatically be granted US citizenship at any time until they reach 18.
After the age of 18, I think there is a different process involved to obtain citizenship.
If the parents were not married, and the father is the US citizen, both father and child may have to take a DNA test to prove paternity.
You can get all of the requirements from the US Citizens section in the US Consulate in Santo Domingo, at the intersection of Maximo Gomez and Av Cesar Nicolas Penson. Bring your passport and your patience, and come early.