Cooper was born in Atlantic City, New Jersey, spent her early childhood in Jacksonville, Florida, and then moved with her family to Princeton in the mid-1930s. She attended Vassar College from 1942 to 1944, and earned a B.A. from the University of Wisconsin in 1946. In 1953–54 Cooper took a year off to get an M.A. at the University of Iowa, where she studied with Robert Lowell, and John Berryman in the Iowa Writers' Workshop.

Cooper joined the faculty of Sarah Lawrence College, in 1950, and remained as a teacher and poet in residence until her retirement in 1987. She held the post of New York State Poet from 1995 to 1997. She died on October 26, 2007, of complications due to Parkinson's Disease.