Stauber made his debut with the Kings during the 1989–90 season, appearing in two games. After two years in the minors, he played in 53 games for Kings between the 1992–93 and 1993–94 seasons. He was traded (along with Alexei Zhitnik, Charlie Huddy, and a draft pick) to the Buffalo Sabres (for Grant Fuhr, Denis Tsygurov, and Philippe Boucher) during the 1994–95 season. Stauber appeared in just one game with the Kings and six games with the Sabres in that season, his last in the NHL. His career NHL stats are 21-23-9 W-L-T, 3.81 GAA, .890 save percentage, and one shutout in 62 games.[4]
Initially hired as a volunteer coach in 1999,[6] Stauber coached at the University of Minnesota's Gophers men's hockey program as their goaltending coach from 2000 to 2008, during which the Gophers won back to back NCAA National Titles in 2002 and 2003.[7]
Stauber has four children from two marriages.[12][13] His son Jaxson is a current goaltender in the NHL.[14] His brother Pete Stauber is a former minor league hockey player and in 2018 was elected US Representative from Minnesota's 8th district.