Koster was a son of Dirk Koster (1855-1935) and Geertje Lakeman. In 1920 he married Bertha Jacoba Gottliebe baroness thoe Schwartzenberg and Hohenlansberg (1891-1993) whom he had also taught.

He was educated in Amsterdam at the Rijksnormaalschool voor Teekenonderwijzers from 1904 to 1906 (where he was taught by Jan Visser and Willem Molkenboer) and the Academy of Visual Arts (Amsterdam) from 1906 to 1910 (where he had Pieter Dupont as a teacher), and in Munich at the Königliche Akademie (with Martin Feuerstein and Carl von Marr as teachers). He mainly painted portraits, self-portraits, cityscapes, coastal views, still lifes and landscapes. Although a painter, he also designed his own house around 1920 at Javalaan 44 in Hilversum.

Koster was a member of various artists' associations: Arti et Amicitiae, Sint Lucas, Artists Association Laren-Blaricum and the Association of Visual Artists (Hilversum).