Inside-forward Baron had played as an amateur for Blyth Shipyard before arriving at Anfield. Due to his young age and diminutive stature he was once refused entry to an away ground with the doorman claiming: 'Don't tell me you're a player. You're only a boy.' He made six league appearances in the 1947/48 season but the side that had won the Championship the previous year was fairly settled and Baron's youth and lack of experience kept him on the sidelines for a while. His big breakthrough came in the 1949/50 season when he only missed four league matches and also played in Liverpool's FA Cup final team against Arsenal at Wembley. In 1951/52 Baron missed just two of the club's 42 league fixtures and scored six goals. A goal every third game in 27 league matches wasn't a bad return in the 1952/53 season.

As Liverpool were relegated in 1954 Baron left for Southend United in Third Division South. He spent four years there scoring 45 goals in 138 league matches. He eventually settled in Suffolk and worked as an insurance agent whilst managing Maldon Town. Kevin Baron died after a long illness in June 1971, only 45 years old. His brother, Gerard Bernard Patrick Baron, who was four years his senior was the oldest victim, aged 67, of the Hillsborough tragedy in 1989.