As a child, Badham was sent to Switzerland to study under Johann Pestalozzi. He afterwards attended Eton College, and in 1830 was elected to a scholarship at Wadham College, Oxford, but only obtained a third class in Classics (1836), a failure which may have been due to the methods of study at Oxford. In 1837 Badham went to Italy, where he occupied himself in the study of ancient manuscripts, in particular those of the Vatican library. He afterwards spent some time in Germany, and was incorporated M.A. at Peterhouse, Cambridge, in 1847. Having taken holy orders, he was appointed headmaster of Louth grammar school, Lincolnshire (1851-1854), and subsequently headmaster of Edgbaston proprietary school, near Birmingham. In the interval he had taken the degree of D.D. at Cambridge (1852). In 1860 he received the honorary degree of doctor of letters at the university of Leiden.
In 1866 he left England to take up the professorship of classics and logic in Sydney University, which he held until his death. He was twice married. Dr Badham's classical attainments were recognized by the most famous European critics, such as CG Cobet, Ludwig Preller, W Dindorf, FW Schneidewin, JAF Meineke, A Ritschl and Tischendorf. Like many schoolmasters who are good scholars and even good teachers, he was not a professional success; and his personality may have stood in the way of his advancement. He remains virtually unknown in the United Kingdom.
He published editions of Euripides, Helena and Iphigenia in Tauris (1851), Ion (1851); Plato's Philebus (1855, 1878); Laches and Eutzydemus (1865), Phaedrus (1851), Symposium (1866) and De Platonis Epistolis (1866). He also contributed to classical periodicals such as Mnemosyne. His Adsortatio ad Discipulos Academiae Sydniensis (1869) contains a number of emendations of Thucydides and other classical authors. He also published some critiques of Shakespeare. A collected edition of his Speeches and Lectures delivered in Australia (Sydney, 1890) contains a memoir by Thomas Butler.