Swenson attended Utah State University in Logan, Utah, graduating in the class of 1934 with a bachelor's degree. She taught poetry as poet-in-residence at Bryn Mawr College, the University of North Carolina at Greensboro, the University of California, Riverside, Purdue University, and Utah State University. From 1959 to 1966 she worked as an manuscript reviewer at New Directions Publishing. Swenson left New Directions Press in 1966 in an effort to focus completely on her own writing. She also served as a Chancellor of the Academy of American Poets from 1980 until her death in 1989. For the last twenty years of her life, she lived in Sea Cliff, New York.
In 1936, Swenson worked as an editor and ghostwriter for a man called "Plat," who became her "boyfriend." "I think I should like to have a son by Plat," she wrote in her diary, "but I would not like to be married to any man, but only be myself."
Her poems were published in Antaeus, The Atlantic Monthly, Carleton Miscellany, The Nation, The New Yorker, The Paris Review, Saturday Review, Parnassus and Poetry. Her poem Question was also published in Stephenie Meyer's book The Host.