Asquith also played on the Heidelberg football and basketball teams.
After returning home from the service in December of 1952, Asquith was planning to continue his education at Ithaca College. But in the spring of in 1953 he received a call from Olean, a farm club of the New York Yankees in the PONY League (now the New York-Penn League). Asquith decided to put college on hold and began his professional baseball career.
Asquith played center field and also pitched. In late August, Asquith injured his rotator cuff and his pitching career was pretty much over at the professional level. After the season he decided to end the hold on college and attended Ithaca.
One of Asquith's teammates at Olean was Bobby Richardson, who went on to win World Series MVP honors in 1960.
While Asquith attended Ithaca, he continued to play baseball in the Buffalo Suburban League, pitching and playing center field. He played for the Cattaraugus Merchants and Jamestown Merchants and later for the Cassadaga Bombers of the County-Grape Belt League from 1961-1964. From 1961 to 1963, the Bombers won 32 straight games.
It was Asquith's accomplishments for more than 20 years of amateur baseball that earned Asquith induction into the Western New York Baseball Hall of Fame in 2009.
After graduating from Ithaca, Asquith taught physical education and coached at Rush-Henrietta High School for two years before arriving at Cassadaga Valley in 1957. He went on to be the head coach for baseball, boys basketball, track and golf. He also served as served as an assistant coach in numerous sports and was the athletic director for four years.
He retired from teaching physical education in 1992, but didn't retire from coaching. He was still the golf coach and the junior varsity girls basketball coach through the 2006-07 school year which gave him a 50-year career of coaching at Cassadaga Valley.
Asquith is also a past officer of the Chautauqua County Athletic Association and was a longtime basketball official. He is a past club champion at Cassadaga Country Club and helped found and is still involved with the Senior Golf Tour.
Asquith was also a member of the National Ski Patrol for 25 years and helped start the high school skiing program at Cockaigne Ski Center.
In addition to the WNY Baseball Hall of Fame, Asquith has also been inducted into the Cassadaga Valley Central School District Athletic Hall of Fame and the Grape Belt Sports Hall of Fame.