Welden's formal art training was taken at Adelphi University in Garden City, New York, where he received his Bachelor of Arts in 1964 and Master of Arts in 1967, after which he studied at the Academy of Arts in Munich, Germany from 1969 to 1971. In 1972 Welden took his first position as a printer at Universal Limited Art Editions as a stone lithographer. There he printed for artists including Jim Dine, Jasper Johns, Robert Motherwell, Robert Rauschenberg and Larry Rivers. In 1977 he joined the faculty of the State University of New York at Stony Brook where he held the position of Assistant Professor of Art until 1985. Since then he has been an Artist in Residence and Assistant Professor of Art at Central Connecticut State University and a faculty member at Suffolk Community College, Long Island University and Southampton College.

In 1972 Welden founded his own press, Hampton Editions Limited, where he has collaborated on prints with Robert Dash, Willem and Elaine DeKooning, Jimmy Ernst, James Brooks, Dan Flavin, Kurt Vonnegut, Alfonso Ossorio and others.