Born in Green Bay, Wisconsin, Crabb received a B.A. from the University of Wisconsin–Madison in 1960 and an LL.B. from the University of Wisconsin Law School in 1962. She was in private practice in Madison, Wisconsin from 1962 to 1968.
After law school graduation, Crabb was a research assistant to George Bunn of the University of Wisconsin Law School from 1968 to 1969, and for the American Bar Association Project on Minimum Standards of Criminal Justice from 1970 to 1971. She served as a U.S. magistrate judge in the Western District of Wisconsin from 1971 to 1979.
On July 21, 1979, Crabb was nominated by President Jimmy Carter to a new seat on the United States District Court for the Western District of Wisconsin created by 92 Stat. 1629. She was confirmed by the United States Senate on October 31, 1979, and received her commission on November 2, 1979. She served as chief judge from 1980-1996.
On March 24, 2010, Crabb took senior status when her successor, William M. Conley, was confirmed as federal judge.
On April 15, 2010, Crabb ruled in a suit that the Freedom From Religion Foundation filed in 2008 against the Obama administration that the National Day of Prayer is unconstitutional. This ruling was unanimously dismissed by a federal appellate court in April 2011 due to lack of standing.
On November 22, 2013 Crabb ruled in another suit the exemption of a housing allowance from the income of clergy was unconstitutional.
On June 6, 2014 Crabb ruled in favor of same-sex marriage, overturning Wisconsin's state passage of a Defense of Marriage Amendment on constitutional grounds including its violation of due process and equal protection.
On November 21, 2016, Judge Crabb joined Circuit Judge Kenneth Francis Ripple in finding that the high number of wasted votes created by the 2011 Wisconsin State Assembly redistricting was unconstitutional partisan gerrymandering, over dissent by District Judge William C. Griesbach.