Arthur was born to Scottish Presbyterian businessman Albert Tyas Harden and Eliza Macalister. His early education was at a private school in Victoria Park run by Dr Ernest Adam. He went to study in 1877 at a Tettenhall College, Staffordshire, and entered Owens College in 1882, now the University of Manchester, in 1882, graduating in 1885. He studied chemistry under Professor Roscoe at Owens College and was influenced by J.B. Cohen.

In 1886 Harden was awarded the Dalton Scholarship in Chemistry and spent a year working with Otto Fischer at Erlangen where he worked on the synthesis of β-nitroso-α-naphthylamine and studied its properties. After receiving a Ph.D. he returned to Manchester as a lecturer and demonstrator and H.B. Dixon and taught along with Sir Philip Hartog. He researched the life and work of Daltonduring these years. In 1895 he wrote a textbook on Practical Organic Chemistry along with F.C. Garrett. Harden continued to work at Manchester until 1897 when he was appointed chemist to the newly founded British Institute of Preventive Medicine, which later became the Lister Institute. He earned the degree Doctor of Science (D.Sc.) from the Victoria University (which included Owens College) in June 1902. Five years later, in 1907 he was appointed Head of the Biochemical Department, a position which he held until his retirement in 1930 (though he continued his scientific work at the Institute after his retirement).

Harden married Georgina Sydney Bridge (died January 1928) in 1900 and they had no children.

Harden was knighted in 1926, and received several honorary doctorates. A Fellow of the Royal Society, he received the Davy Medal in 1935.