Brower has been a judge of the Iran–United States Claims Tribunal since 1983.
From 1961–1968, Brower was a lawyer with the law firm of White & Case LLP.
From 1969–1973, he was Acting Legal Adviser to the United States Department of State.
He then returned to White & Case and co-founded its Washington, D.C., office.
Brower has been a Visiting Fellow at Jesus College, Cambridge and a distinguished visiting professor at the University of Virginia Law School.