After receiving his high-school diploma, he studied law and economics at the University of Hamburg, where he graduated in 1969.

Four years later, in 1974, he was elected as a member of the Hamburg Parliament (German: Hamburgische Bürgerschaft).

Voscherau became First Mayor of the Free and Hanseatic City Hamburg on 8 June 1988, and held this office until 8 October 1997.

Although he stepped down from active politics in Hamburg that year, and started working as notary again, he remained vice president of the SPD until 2001.

His father was Carl Voscherau, a German stage and film actor, his brother Eggert is member of the board of directors of BASF.