Born near Airdrie in Lanarkshire, Stevenson was educated at Caldercruix Advanced School before joining the British Army.
[1] In 1951, Stevenson was elected as president of his local branch of NUBSO, and ten years later he began working full-time for the union, based in Street.
[1][2] As leader of NUBSO, Stevenson focused on a campaign to reduce the length of the working week, and for European Economic Community policy to support British shoemaking.
He promoted education among trade union members, and served on the council of Ruskin College.
[1][2] Faced with declining membership, Stevenson negotiated a merger between NUFLAT and the National Union of Hosiery and Knitwear Workers which formed the National Union of Knitwear, Footwear and Apparel Trades; he retired at the end of 1990, as the merger was effected.