Born in West Derby, Liverpool, Lancashire, Balmer came from a family of footballers: Billy and Bob both played for Everton during the early part of the 20th century, whilst uncle Jack was an amateur at Goodison Park.
Balmer played for Collegiate Old Boys and Everton, where he was an amateur, before he made the short journey across Stanley Park and was signed by Liverpool manager George Patterson, who gave him his first professional contract on 23 August 1935.
The 19-year-old made his debut on 21 September 1935 at Elland Road against Leeds United in a Football League First Division match, however, it wasn't a success as Liverpool lost 1–0.
Upon resumption of the national competitions, Balmer formed a partnership with Albert Stubbins; they both finished the first post-war season (1946–47) with 24 goals, helping to guide Liverpool to their first championship in 24 years.
It was during the war that he gained his only international recognition, playing for England in a wartime match against Wales at the Racecourse Ground in Wrexham in front of a crowd of 17,000 on 18 November 1939.