Born Edward Walter Kriaučiūnas in Chicago to Lithuanian immigrant parents, Krause grew up in the Town of Lake Section or, as it was once known as, Back of the Yards. His brother, Felix Krause, was the captain of the Lithuania national basketball team in 1937. His surname was shortened to Krause by his high school football coach, who couldn't pronounce Kriaučiūnas (Lithuanian pronunciation: [kreo-chunas]).
At Notre Dame, Krause competed in track, baseball, football and basketball, becoming the first Notre Dame player to make Hall of Fame in both basketball and football. In basketball, he was a three time Consensus All-American (1932–1934). Krause played football for the Fighting Irish under Knute Rockne. He graduated cum laude from Notre Dame in 1934 with a journalism degree.
Krause's coaching career included a five-year stint as head coach in all sports at Saint Mary's College in Winona, Minnesota, now Saint Mary's University of Minnesota; an assistant football coach at the College of the Holy Cross and the University of Notre Dame for ten years; and head basketball coach at the University of Notre Dame in 1943 and again from 1946 to 1951, when he compiled a record of 98–48 (.671). As acting head football coach at Notre Dame, filling in for an ailing Frank Leahy, Krause was 3–0.
Krause served in the United States Marines during World War II including a 14-month stretch as an air combat intelligence officer in the South Pacific. Krause became the assistant athletic director at Notre Dame in 1948. In March 1949, he was named athletic director, succeeding Frank Leahy who stepped down from the role to focus on his post as head football coach.