Spencer appeared on Broadway in In the Boom Boom Room and Execution of Justice for which he was a Drama League Award honoree. He won an Obie Award for his role in Still Life and received a Drama Desk nomination for his performance in The Day Room off-Broadway. His other New York stage credits include Fishing, The Ballad of Soapy Smith, El Salvador, Frankie and Johnny in the Claire de Lune, Division Street, Good As New, The Exonerated and Glimmer, Glimmer and Shine. Regional credits include mountings of Lakeboat, Red River, Loose Ends, This Story of Yours and The Glass Menagerie.
Spencer had homes in both New York and Los Angeles.