Born in Dundee, Arthur started his career with North End, had a short spell with Portsmouth, before joining Falkirk in 1934. He had fallen out of the starting line-up when Dundee F.C. made enquiries in February 1936. His debut was against Aberdeen, in the Second Eleven Cup, at Pittodrie the following Saturday. He hit the ground running, becoming a prolific goal scorer.
A loss of form, towards the end of the 1937-1938 season saw his position under threat. However, although the last to put pen to paper, he did resign for the following term.
1938-1939 was proving to be a better season, and Cardiff City were showing interest in Arthur, but they were beaten to his signature by Barnsley, who were top of the English Third Division, in December. Their manager had travelled up to Forfar to watch him play, and offered the Dundee officials £1,000.
Baxter returned to Dundee at the outbreak of war to take up “war work.” During this time he played for Dundee United, playing in the Scottish war-time cup.
Arthur enlisted as a Private in the Gordon Highlanders, attached to the 1stLondon Scottish regiment in 1941, He was initially stationed near Aberdeen, and represented the S.F.A. and Aberdeenshire Select. On being transferred to the south of England, an opportunity arose to play for Charlton Athletic. On his return up north he got a chance to play for the Dons
After training he headed overseas with the Central Mediterranean forces. Arthur died at the Battle of Rimini, 5th September 1944, he had previously been wounded in March. He is buried at Gradara War Cemetery, Italy. Leaving behind a widow, and a son, who resided at 87 Strathmartine Road.