 Osborn served as president of the Metropolitan Museum of Art from 1941 to 1947 and sat on its board of trustees for forty-five years.On June 3, 1886, Osborn was married to philanthropist and social reformer Alice Clinton Hoadley Dodge (1865–1946).Alice was a daughter of William E. Dodge, Jr. and the younger sister of Grace Hoadley Dodge, William E. Dodge III, and Cleveland Hoadley Dodge.