Baker became a prolific producer of engraved maps from the late eighteenth century onwards, including for William Faden. As a map engraver and publisher, he resided first at 32 High Street, Islington from 1791–1793 and then in Lower Street, Islington from 1798–1800. In 1804, he became principal engraver to the Ordnance Survey and he and his firm were regarded as “the best topographical engravers in Europe” (Seymour). There he worked with his three sons: Benjamin Richard (1792–1876), one of the initial surveyors who did the Survey of Ireland from 1824–1845; Alfred (1797–1878); and George (1804–1876).
From a long line of instrument-makers, including his grandfather Edward (1705–1779) and his uncle Benjamin (1737–1821), the young Benjamin was initially apprenticed to watchmaker, Thomas Beresford, in 1782, but soon switched allegiance to map-engraver William Palmer. His great grandmother was the daughter of Benjamin Cole (1667–1729), an engraver and bookbinder of Oxford. She was also the younger sister of Benjamin Cole (1695–1766), mathematical instrument-maker of Fleet Street, London. His elder sister, Mary, married John Newton of the globe-making family; and his eldest brother, Edward (1757-before 1836), became a mathematical-instrument-maker of New Street, Fetter Lane.