Geske was born in Port Washington, Wisconsin and raised in Cedarburg. She received her bachelor's and master's degrees from Beloit College in 1971 and her law degree from Marquette University Law School in 1975.
Geske became the chief staff attorney for the Legal Aid Society of Milwaukee, where she worked until 1979. She was an assistant professor at Marquette University Law School from 1978 to 1981 and was the director of the Marquette University Law School's Clinic for the Elderly, which she founded. Beginning in 1981, Geske served in the Milwaukee County circuit court for 12 years, until she was appointed to the Wisconsin Supreme Court by Governor Tommy Thompson in 1993. She won election to the court in 1994, and resigned from the bench in 1998. From February to May 2002, she served as the interim county executive for Milwaukee County.
She held the Association of Marquette University Women's Chair in Humanistic Studies in 2000 and 2001, and she is on the faculty of the National Judicial College in Reno, Nevada. Among her numerous awards, in both 1994 and 2002 the Milwaukee Bar honored Geske with its Lawyer of the Year award.