Brams was born Concord, New Hampshire in 1940. He earned his B.S at Massachusetts Institute of Technology in Politics, Economics, and Science in 1962. He went on to earn his Ph.D in Political Science in 1966.
He worked briefly in U.S. federal government positions before taking an Associate Professor position at Syracuse University in 1967. He moved to New York University in 1969, where he is a Professor in the Department of Politics.