He was born at Lincoln, England, 21 October 1768, the son of the Rev. Henry Beste, D.D., prebendary of Lincoln Cathedral. His mother, Magdalen, daughter and heiress of Kenelm Digby, Esq., of North Luffenham in Rutland, claimed to be the representative of the extinct male line of the historic Sir Everard and Sir Kenelm Digby. His father dying in 1782, Henry was sent two years later by his mother to Oxford. He became a commoner of Magdalen College, where he took his B.A. degree in 1788 and his M.A. in 1781. He was afterward elected to a fellowship, which he resigned when the family estates came to him on the death of his mother. In September, 1791, he took deacon's orders in the Anglican church, and a little later retired to Lincoln, where he was active as a preacher. Doubts about the spiritual authority of the Established Church sprang up in his mind, which were strengthened by intercourse with the Abbé Beaumont, then in charge of the small Catholic chapel at Lincoln. The result was that he was received into the Catholic Church by Rev. Mr. Hodgson, Vicar-General of the London district, 26 May 1798. He died at Brighton, 28 May 1836. In 1800, he had married Sarah, daughter of Edward Sealy, Esq., and was the father of the author John Richard Digby Beste.
His first works were a treatise entitled "The Christian Religion briefly defended against the Philosophers and Republicans of France (octavo, 1793), and in the same year a discourse on "Priestly Absolution" which was republished in 1874. This latter work anticipated some of the Tractarian arguments and met with the commendation of members of the University of Oxford in 1794. After his conversion Beste was an occasional contributor to Catholic periodicals. He also travelled abroad and spent several years in France and Italy. Cardinal Wiseman met him at Rome in the Jubilee of 1825, and mentions him in his "Last Four Popes" (Boston, 1858), p. 245). In 1826 Best English Family's Residence there during that period, preceded by some account of the Conversion of the Author to the Catholic Faith" (octavo). Two years later he wrote a similar book on his stay in Italy. Ten years after his death appeared his last work, called "Poverty and the Baronet's Family, a Catholic Story" (12mo, 1846).