He was orphaned and raised in New York by his aunt and uncle, the parents of songwriter, George M. Cohan.  Mainly self-taught, Costigan studied briefly at the Art Students League and worked as a lithographer of theater posters.  From that job he learned the printing processes that he later used in his many etchings.  He began making his name in the fine arts in 1920, and throughout the decade, reaped numerous important prizes for his oils, watercolors and prints.

He served in the infantry in World War I and supported himself during the Depression with magazine illustration.  For about a year during World War II, to remain financially solvent, he worked the night shift as a machine operator in a defense plant while continuing to paint and etch by day.

In the late 1960s, he enjoyed a revival of interest in his fine arts talents when more than 50 pieces were borrowed from museums and private collections to be toured nationally in a Smithsonian-sponsored Costigan retrospective.