Crosby served in the Nebraska legislature, as Speaker of the Nebraska legislature, and as Lieutenant Governor (1947-1949) before being elected governor. Crosby was known as "The Boy Governor from North Platte."
In April 1954, Governor Crosby appointed Eva Bowring to the U.S. Senate to fill out the term of the late Dwight Griswold. In July 1954, Governor Crosby appointed Samuel Williams Reynolds to the U.S. Senate to fill out the term of the late Hugh A. Butler.
Crosby was a candidate for the US Senate in the Republican primary in 1954. He was defeated by Carl T. Curtis. U.S. Senator Hazel H. Abel, who had been elected to serve the last two month of the term of the late Dwight Griswold, resigned a few days before the end of her term, and Governor Crosby appointed Carl T. Curtis to serve the remainder of her term. That gave Senator Curtis slightly more senority than other senators elected at the same time.
Following his service as governor, Crosby practiced law in Lincoln, NE, until shortly before his death. He died on January 7, 2000. He is buried in Lincoln Memorial Park, Lincoln, NE.