Born in Utica, New York, he debuted as a singer in Scranton, Pennsylvania, as the Conte di Luna in Il trovatore, in 1940. As director, his first production was of Die Fledermaus, for the Pittsburgh Opera, in 1944.
Alexander sang with companies in Latin and North America and Europe, including Bayreuth (Sixtus Beckmesser in Wieland Wagner's Die Meistersinger, 1963 and 1964). Operas in which he appeared include Lulu, Il prigioniero, Moses und Aron, Der fliegende Holländer, Lohengrin, Parsifal (as Amfortas), Der Ring des Nibelungen (as Wotan), and Tristan und Isolde.
In 1947, the baritone sang in Ariadne auf Naxos (as the Major-Domo) and Salome (as Jochanaan) at the New York City Opera.
His 1950 performance, at Carnegie Hall, of Mahler's Eighth Symphony, under Leopold Stokowski, was recorded. In 1968, he portrayed Dr Schön in the Stuttgart film of Wieland's production of Lulu, opposite Anja Silja.
As director, Alexander's productions were seen in Toronto, Stuttgart, Chicago, New Orleans (Les contes d'Hoffmann, 1947), Pittsburgh, and Santa Fe. His stagings included Fidelio, The Rape of Lucretia, Madama Butterfly, Salome, and Der fliegende Holländer.
Of the Kammersänger's Wotan, Wieland Wagner declared that, "No vocal giant has moved me so deeply in Wotan's Farewell as Alexander."