Downs was born on 4 July 1893 in Kingston upon Hull.^[3] He was educated at Abbotsholme School, then an all-boys private boarding school in Uttoxeter, Staffordshire.^[1] In 1912, he matriculated into Christ's College, Cambridge, having been awarded a scholarship.^[3] In 1915, he graduated Bachelor of Arts (BA) with first-class honours in Modern and Medieval Languages (German, Old Norse and Anglo-Saxon).^[1]

In 1918, Downs was appointed a college lecturer in English and modern languages at Christ's College, Cambridge.^[1] In the summer of 1918, with hostilities still going on, Downs held a meeting in his rooms leading to the foundation of the Modern Humanities Research Association, dedicated to the study of European languages.^[4] The following year, in 1919, he was elected a Fellow of the college.^[3] From 1920, he was also a university lecturer in English and modern languages. In 1928, he was appointed a college tutor.^[3] He served as a Member of the Cambridge University Council, the governing body of the University of Cambridge, from 1939 to 1944.^[1] In 1947, he became Senior Tutor and Vice-Master of Christ's.^[3] In 1950, he was appointed the 31st Master of Christ's College, Cambridge.^[5] The same year, the Chair of Scandinavian Studies was established and he was elected to the position becoming a professor.^[2] From 1955 to 1957, served as the then part-time Vice-Chancellor of the University of Cambridge.^[1] He retired in 1963.^[1]

Downs remained in Cambridge following his retirement and lived on Grange Road for a number of years. He later moved into the Hope Nursing Home in Cambridge and died there on 3 March 1984.^[1] A diabetic, he had contracted bronchopneumonia which proved fatal.^[1] His memorial service was held in the chapel of Christ's College, Cambridge, on 2 June 1984.^[6]

On 1 July 1924, Downs married Elsie Ada Maud Drew (1887–1965). She was a fellow of Girton College, Cambridge, where she tutored in English literature.^[1] Together they had a son, Jonathan Downs, in 1930. Mother and son moved to the United States of America during World War II but, at the age of nine, Jonathan was hit and killed by a truck while outside his home in Northampton, Massachusetts.^[1] Their marriage subsequently broke down and they divorced. On 14 September 1946, Downs married for a second time to Evelyn Faith Marion Wrangham (née Doubble) (1907–1977). She was a widow who had worked as his secretary before their marriage.^[1]

Downs was awarded an honorary Doctor of Letters (DLitt) degree by the University of Hull.^[1] He was appointed Commander of the Order of the Polar Star (KNO) by Sweden in 1954 and Officer of the Legion of Honour by France.^[3]