Although he spent most of his career in New York City, Alexander occasionally traveled to perform as a guest artist with many of the world's leading opera houses, both in the United States and Europe.
His signature roles included Alfredo in La traviata, Arbace in Idomeneo, Belmonte in Die Entführung aus dem Serail, the Duke of Mantua in Rigoletto, Edgardo from Lucia di Lammermoor, Ferrando in Così fan tutte, Hoffman in Les contes d'Hoffmann, Lieutenant B. F. Pinkerton in Madama Butterfly, Rodolfo in La bohème, and the title roles in Don Carlos and Faust.
He traveled widely with the company during the Fall of 1951, notably portraying Alfredo in Giuseppe Verdi's La traviata opposite a young Beverly Sills as Violetta.
The following fall he once again traveled with the Charles Wagner Opera Company, this time singing Don José in Bizet's Carmen opposite Sill's Micaëla.
In 1957 Alexander made his debut with the Philadelphia Grand Opera Company as Lieutenant Pinkerton in Puccini's Madama Butterfly opposite Dorothy Kirsten as Cio-Cio San.
The following year he returned to the New York City Opera to portray the role of Henry Morosus for the United States premiere of Richard Strauss's Die schweigsame Frau.
He continued to perform periodically at the New York City Opera up through the 1970s, notably excelling as Walther in Wagner's Die Meistersinger and as Bacchus in Richard Strauss's Ariadne auf Naxos.
He made his first appearance on the Met stage as Ferrando in Così fan tutte on December 19, 1961, with Teresa Stich-Randall as Fiordiligi, Rosalind Elias as Dorabella, Theodor Uppman as Guglielmo, Roberta Peters as Despina, and Frank Guarrera as Don Alfonso.
His discography includes recordings of Bellini's Norma (with Dame Joan Sutherland and Marilyn Horne, 1964) and Donizetti's Anna Bolena (opposite Elena Souliotis, 1968–69).