Born in Utica, New York, on May 7, 1880, Coxe received an Artium Baccalaureus degree from Yale University in 1901 then attended Cornell Law School.

He was confirmed by the United States Senate on April 29, 1929, and received his commission on May 1, 1929.

His service terminated on December 21, 1957, due to his death in Old Lyme, Connecticut.

Coxe was the son of Alfred Conkling Coxe Sr., who was a judge on the United States Court of Appeals for the Second Circuit, and the great-grandson of Alfred Conkling, who served as a United States representative from upstate New York and a judge on the United States District Court for the Northern District of New York.

[citation needed] He was also a grand nephew of Roscoe Conkling, who was a Congressman and Senator from New York and boss of the state's Republican political machine.