Draper, the son of the distinguished physician and chemist John W. Draper, was born in Prince Edward County, Virginia. He studied at the City University of New York, completing the course in medicine in 1857 before he was old enough to graduate. He obtained his MD in 1858, spending the preceding months in Europe where his interest in astronomy was aroused by a visit to the observatory of the third earl of Rosse at Parsonstown, Ireland. On his return to New York he joined the Bellevue Hospital and was later appointed professor of natural science at the City University in 1860. Draper later held chairs of physiology (1866–73) and analytical chemistry (1870–82) and in 1882 succeeded his father briefly as professor of chemistry. He retired from the university in 1882 in order to devote himself to astronomical research but died prematurely soon after.

One of the most important events in Draper's life was his marriage in 1867 to Anna Palmer, daughter and heiress to Courtlandt Palmer who had made a fortune in hardware and New York real estate. His wife's money allowed him to purchase a 28-inch (71-cm) reflecting telescope and to begin a 15-year research partnership.

Draper was interested in the application of the new technique of photography to astronomy. He started by making daguerrotypes of the Sun and Moon but in 1872 succeeded for the first time in obtaining a photograph of a stellar spectrum, that of Vega. In 1879 he found that dry photographic plates had been developed and that these were more sensitive and convenient than wet collodion. By 1882 he had obtained photographs of over a hundred stellar spectra plus spectra of the Moon, Mars, Jupiter, and the Orion nebula. He also succeeded in directly photographing the Orion nebula, first with a 50-minute exposure in 1880 and then, using a more accurate clock-driven telescope, with a 140-minute exposure. He thus helped to establish photographic astronomy as an important means of studying the heavens.

At the time of his death his widow hoped to continue his work herself, but with prompting from Edward Pickering at the Harvard College Observatory, she set up the Henry Draper Memorial Fund. It was with the aid of this fund that the famous Henry Draper Catalogue, some nine volumes with details of the spectra of 225,000 stars, was published from 1918 to 1924 through the labors of Pickering and Annie Cannon.