William Baxter Palmer Closson was born in 1848 in Thetford, VT. He was educated at the Thetford Academy.
Closson worked as a clerk in the railroad office before moving to Boston, where he worked as an apprentice wood engraver with Samuel S. Kilburn. In Boston, he studied drawing at the Lowell Institute.
Before taking up painting full time, Closson worked as an engraver for Harper's and other Boston-area book publishers. He transitioned to painting in 1890.
In addition to The National Arts Club, his work can be found in the collections of the Smithsonian Institution, the Wadsworth Atheneum Museum of Art, and the Cleveland Museum of Art.