Stevens was born in Brockton, Massachusetts, but grew up in Pembroke, Massachusetts. As a youth, he played in the 1978 Quebec International Pee-Wee Hockey Tournament with the Hobomock minor ice hockey team from Pembroke. While attending Silver Lake Regional High School in Kingston, Massachusetts, Kevin Stevens played both hockey and baseball. He was invited to try out for both the Toronto Blue Jays and the Philadelphia Phillies; however, accepting that he was not a great hitter in baseball, he decided to play hockey instead. Stevens accepted a full scholarship to play hockey for Boston College, and was drafted in the sixth round (108th overall) in the 1983 NHL draft by the Los Angeles Kings. Several months later, his rights were traded to the Pittsburgh Penguins for Anders Håkansson, a left winger from Sweden who had recently been traded to the Penguins along with Ron Meighan from the Minnesota North Stars for the Penguins' first round pick in the 1983 draft (Brian Lawton).