Born in Cremorne, a North Shore suburb of Sydney, New South Wales, Border grew up with three brothers in the nearby suburb of Mosman. His father John, from Coonamble in rural New South Wales, was a wool classer and his mother Sheila was the proprietor of a corner store. The family had a spacious backyard for playing games, and Mosman Oval, the home of district cricket and baseball clubs, was across the street. Border attended North Sydney Boys High School, and earned his leaving certificate in 1972.

Throughout his early years, Border played in cricket teams two or three years older than his age group. He also played for Mosman Baseball Club, where he developed his fielding and horizontal-bat shots. Aged sixteen, he made his début for Mosman in Sydney Grade Cricket as a left arm orthodox spinner and batted at number nine. He won selection for the 1972–73 Combined High Schools team in the intrastate carnival. During this time, he was coached by Barry Knight, a former England international.

Border accumulated more than 600 runs in grade cricket in 1975–76, and at the start of the following season, he made two consecutive centuries to earn selection for NSW. In the absence of a number of Test players, Border made his debut against Queensland at the SCG in January 1977. He compiled 36 and took the last three catches of the match, as his team claimed victory. Border resigned from his job as a clerk in the film library of BHP to spend the 1977 English season playing for Downend in the Gloucestershire Western League. The highlight of his stay was 159 not out in an invitational match against Cambridge University. In Australia, Border compiled 617 runs at 36.29 average during the 1977–78 Sheffield Shield season. He then returned to England and played for East Lancashire Cricket Club in the Lancashire League, scoring 1191 runs at 56.71 and taking 54 wickets at 18.60.