A. Jasper Anderson studied at Magdalen College, University of Oxford, and St Bartholomew's Hospital, London. He was admitted as a member of the Royal College of Surgeons of London (MRCS) in 1882. That same year he married Miss S.R. Cooke, with whom he had a son and a daughter. In 1883 he qualified as Bachelor of Medicine (MB). He also qualified as Master of Arts (MA), and obtained the Diploma in Public Health at Cambridge in 1888. Having won the Radcliffe Travelling Fellowship at Oxford he continued his studies at the Universities of Leipzig and Vienna. After a period of general practice in England he took up public health work and became medical officer of health for Blackpool for eleven years. During this period he published an article on 'The past, present and future of the medical officer of health' in the journal Public Health (1894).
In October 1901 Anderson assumed duty as the first full-time Medical Officer of Health of Cape Town, succeeding Dr E.B. Fuller*, who had served in a part-time capacity until he resigned the post in 1899. Anderson was licensed to practice in the Cape Colony on 15 November 1901. He held his post until his retirement, owing to ill health, at the end of 1922. In 1909 he proposed to present a series of lectures on hygiene at the South African College, forming a six-month course that was recognised by the Royal College of Physicians and the Royal College of Surgeons of Edinburgh, and by the Faculty of Physicians and Surgeons at Glasgow. The proposal was approved by the College Council, provided there was enough student interest. After his retirement he became professor of hygiene at the University of Cape Town until 1923.
Anderson was an active member of the British Medical Association and was elected president of its Cape of Good Hope Western Province branch in 1909. He became a member of the South African Philosophical Society almost immediately upon his arrival in 1901 and remained a member until the Society changed into the Royal Society of South Africa in 1908, when he was elected one of its founding fellows, and served on the society's council during 1917-1918. In 1903 he also joined the South African Association for the Advancement of Science, serving on its council for 1912/13. He was a member of the Sanitary Institute of London, acted as a local examiner for the institute, and published various papers on public health and sanitation.