Henry Babington Smith was born on 29 January 1863, the fifth surviving son of Archibald Smith, a mathematician, and Susan Emma Parker, daughter of Mary Babington and Sir James Parker. He was educated at Eton and Trinity College, Cambridge, where he took a BA in 1886 with first classes in parts I and II of the Classical Tripos, and was elected a Fellow of the College in 1889.
On leaving Trinity, Smith embarked on a distinguished career as a civil servant. He held a number of posts, principally of a financial nature, which involved service overseas in India, South Africa, Turkey, and, briefly, the USA, besides a spell at home in the Treasury. He became administrator in the National Bank of Turkey in 1909, was Chairman of the Indian Currency and Finance Commission in the 1920s and a Director of the Bank of England in 1920 He also spent some years in the General Post Office and later chaired several important tribunals and Royal commissions. He was awarded the KCB in 1908.
During the early part of his career, Smith acted as private secretary to the 9th Earl of Elgin, then Viceroy of India. In 1898 he married Elgin's daughter, Lady Elisabeth Bruce. He died in 1923.