Coster was born in Amsterdam. On February 26, 1919, he married Lina Maria Wijsman, who held a degree in Oriental languages. Eventually, she was one of the first women to obtain a doctorate degree in this field from the University of Leiden. Dirk and Miep had two sons and two daughters (Hendrik, Ada, Els, and Herman). Coster is known as the co-discoverer of hafnium (element 72) in 1923, along with George de Hevesy, by means of X-ray spectroscopic analysis of zirconium ore. The discovery took place in Copenhagen, Denmark. Its name is derived from "Hafnia", the Latin name for Copenhagen.