Anderson was born in 1877 to James George Skelton Anderson and Elizabeth Garrett Anderson. Anderson's father was a shipping magnate who merged the family shipping business, Anderson, Anderson & Co., with Frederick Green & Co. on 12 February 1878 to create the Orient Steam Navigation Company. Anderson's mother was the first British woman in England to qualify as a doctor. He was one of three children born to the couple. One of his sisters, Louisa Garrett Anderson, followed in her mother's footsteps and became a doctor herself, serving during World War I as the head of a military hospital, while Anderson joined his father in the family's shipping enterprise in 1897. Prior to joining the company, Anderson was educated at Eton College and Trinity College, Oxford. Once established in the shipping industry, Anderson expanded into the related field of rail transport, becoming director of Midland Railway in 1911, a seat he maintained through the merger of that railway in 1923 into London, Midland and Scottish Railway.
Anderson utilised his business experiences internationally in service of the government during the First World War. Appointed vice-chairman of the Royal Commission responsible for regulating the distribution of wheat supplies, he dealt generally with Great Britain's western allies. He dealt specifically with the United States first in processing enemy cargo aboard US vessels under Walter Runciman and, after the US entered the war, in marketing wheat in the United States as well as Canada along with Arthur Balfour. In the summer of 1917, Anderson was appointed to replace Eric Geddes as Admiralty controller, a position that made him one of the Lords Commissioners of the Admiralty. During Geddes' term, the Controller had been put in charge of overseeing construction and repairs of both the Royal and Merchant navies, which were taking a toll from submarine attacks. Geddes had set very high quotas for production that the Controller's office, even with delegation, was unable to meet both from shortage of materials and manpower. Tensions were high on both sides, with Joseph Davies specifying his irritation with Anderson, while Anderson, in response to one notable shortage of shipbuilders in November 1917, chided the military for "a complete lack of sense of proportion", suggesting that recruitment would be more successfully accomplished by appealing to wives and union officials in towns known for shipbuilding than seeking the Home Office. Anderson resigned from the position in 1918. In spite of these challenges, Anderson was widely honoured. In 1917, he was appointed a Knight Commander of the Order of the British Empire, an Officer of the Légion d'honneur, and an Officer of the Order of the Crown of Italy in reward for his wartime services.
On 9 June 1903, Anderson wed Muriel Ivy Duncan of Surrey. The couple parented four children, including Sir Donald Forsyth Anderson who followed his father into the shipping industry. He died on 4 May 1952.