The Montreal native put up significant offensive numbers in junior with the Verdun Epeviers and Sorel Black Hawks of the QMJHL. Kasper topped the 100-point mark twice and scored 57 goals in 1979-80. He was chosen 81st overall by the Boston Bruins in 1980.

After beating out top draft pick Barry Pederson in training camp, the hard working centre played 76 games as a rookie in 1980-81, scoring 56 points and demonstrating unusual poise for a young player. He ended up reaching the 20-goal mark four times in Boston while emerging as one of the best checking centres in the game. He formed an effective defensive troika with Keith Crowder and veteran Wayne Cashman and was often a threat to score while shorthanded. On January 9, 1985 he scored two short-handed goals while killing the same penalty against Toronto during a 5-3 win at Maple Leaf Gardens.

Kasper's proficient two-way play was rewarded when he won the Frank J. Selke Award in 1982. His excellent work helped the Bruins reach the semi-finals in 1983 and, in 1988, he scored seven goals while helping the reach the final. In 1988-89 the Bruins traded Kasper to the L.A. Kings form homegrown Bobby Carpenter. "The Friendly Ghost" excelled on the West Coast and was a fine third line centre behind Wayne Gretzky and Bernie Nicholls. Kasper later played briefly for the Philadelphia Flyers and Tampa Bay Lightning before retiring in 1993 after taking part in 821 regular season games and 94 post-season matches.

Kasper returned to hockey in Boston as an assistant coach to Brian Sutter. On July 22, 1994 he was named the head coach of the Providence Bruins of the AHL and led the team to the second round of the Calder Cup playoffs. On May 25, 1995 he succeeded Sutter as the Bruins' bench boss and led the club to a 91-point season. After a first round playoff ouster at the hands of the Florida Panthers and a disappointing 26 wins in 1996-97, Kasper was replaced by Pat Burns.