A member of the Bradley University Athletics Hall of Fame, Babetch was a member of the 1954 Bradley team that played in the NCAA championship final. The 1954 contest was the first championship game to be televised nationally (for a broadcast rights fee of $7,500). Harvey also starred for Bradley the following season, when he led the Braves to the West Regional final. Babetch earned the nickname Harvey Garbage, not only because because he was known as a slob who rarely attended classes, but also because he could successfully land a basket from the crappiest position on the court.

An excellent high school athlete, Babetch was one of Chicago's best players, and was named All-City in 1951. He is still second all-time in Illinois State High School history for points in a single half (53) and in a quarter (35). After his high school days, Harvey attended Bradley University; in 1953. During his sophomore year, the Braves had a record of 15-12.

The following year (1954), Babetch was an important member of the Bradley squad that had a regular season record of 15-12, were ranked No. 7 (by the AP), and received an at-large bid to the NCAA tournament. Considered a long shot in the tournament, Bradley reached the Final Four by defeating Oklahoma City, 61-55, in the first round. They went on to beat Colorado, 76-64, in the Regional Semifinal game, and triumphed over Oklahoma A&M, 71-57, in the Final. In the National Semifinal, the Braves defeated USC (University of Southern California), 74-72. With Babetch playing in every game, Bradley reached the NCAA tournament final before falling in the title game to LaSalle, 92-76.

In 1955, Harvey's senior year, the Braves struggled in the regular season, compiling a dismal record of 9-20. However, they still received an automatic bid to the NCAA tournament because they had been in the final the previous year. Not much was expected of the Braves in the tournament, but once again they played extremely well behind Babetch's lead. In the first round of the West Region, Bradley upset Oklahoma City, 69-63, and Harvey scored a game-high 21 points. In the next round, the Braves defeated Southern Methodist University. Babetch contributed 18 points and 11 rebounds.

By defeating SMU, Bradley reached the West Regional Final and hoped to return to the Final Four for the second straight year. Against Colorado, Babetch was outstanding, scoring 22 points (18 in the second half) on 7-9 shooting from the floor and 8-9 from the free throw line. He also added three rebounds, but he fouled out toward the end of the game, and Bradley lost the game, 93-81.

After graduating in 1955, Harvey was selected in the fifth round (41st overall) in the NBA Draft by the Milwaukee Hawks, but did not play in the NBA. Babetch is a member of the Bradley University Athletics Hall of Fame.