Crowe was born in Henderson, a suburb of Auckland, to a family of cricketers. His father, Dave Crowe, played first-class cricket for Canterbury and Wellington and his older brother, Jeff Crowe, played Test cricket. One of the brothers' first cousins is actor Russell Crowe. In 1968, Martin Crowe joined his father and brother at the Cornwall Cricket Club, with which he maintained a lifelong connection. At Auckland Grammar School, which he attended from 1976 to 1980, he was deputy head boy in his final year. He captained the school's cricket team, and also played rugby union, as a wing. He is also the great grandson of All Black Francis Jervis (his mother's maternal grandfather).

Crowe made his first-class debut in January 1980, playing for Auckland against Canterbury. Aged 17 at the time, he scored 51 runs in his first innings, which was his team's highest score. In 1981, having been named New Zealand's Young Cricketer of the Year, Crowe was given the opportunity to spend six months on the ground staff of Lord's, simultaneously playing for the Marylebone Cricket Club. Returning to New Zealand for the 1981–82 domestic season, he scored a maiden first-class century, making 150 runs against Canterbury. Crowe switched to Central Districts after the 1982–83 season. His career for Central Districts was limited by his international duties, but in 32 first-class appearances (from 1983 to 1990), he averaged 68.72, with 13 centuries. Crowe's highest score for the team (and in all New Zealand domestic cricket) was 242, made against Otago in January 1990. That season was his last for Central Districts, as he transferred to Wellington prior to the 1990–91 season. In five seasons