Born in Halcott Centre, New York, Crosby attended Yale University beginning in 1889; while there, he was a popular student and sportsman.

During his junior year, he was replaced on the football team by freshman Frank Hinkey and never returned to a starting position.

Crosby was invited in 1892 to serve as head coach of the United States Naval Academy football program.

He accepted the position, and, using unusually rigorous practicing strategies, led the team to a 5–2 record, culminating in an upset victory over rival Army in the Army–Navy Game.

Following the season's conclusion, Crosby returned to New York to continue his studies, but he was hospitalized after an illness he contracted while coaching worsened shortly after his arrival.

Crosby was popular while at Yale, and was a member of both Delta Kappa Epsilon fraternity and the secret undergraduate society Skull and Bones.

While observing a team practice, Hartwell and Crosby noticed the play of backup end Frank Hinkey.

After watching Hinkey for a time, Hartwell decided to return to practicing for fear of losing his starting position; Crosby did not consider this necessary.

[14][15] The following week, on October 22, Crosby achieved his first coaching win when Navy defeated the Lafayette Leopards 22–4.

This was followed by a victory the next week over the small Franklin & Marshall College; Navy won in a 24–0 shutout, their largest such win since the 1890 Army–Navy Game.

[19][20] Crosby's actions proved effective as, on November 26 at West Point, Navy led Army for the entire game and won easily, 12–4.

He later became one of the longest-serving members of the college football Rules Committee, helping to legalize the forward pass and ban the flying wedge, among other things.