He came to America with his father, Dr. Leopold Damrosch, in 1871, having already studied music under Dionys Pruckner and Vogt. He studied in New York under Von Inten and his father. He also studied in Europe under Moritz Moszkowski.
He originally intended to adopt a business career, and to that end went to Denver, Colorado, but the musical impulse proved too strong, and in 1884 he was an organist, conductor of the Denver Chorus Club, and supervisor of music in the public schools.
For some years he was chorusmaster at the Metropolitan Opera House in New York. He has also conducted the Mendelssohn Glee Club from 1885 to 1887 and other important organizations. In 1892 he organized the People's Singing Classes, which has since developed into the People's Choral Union, with a membership of 1,200, and he was also instrumental in founding the Musical Art Society of New York.
In 1897 he became supervisor of music in the public schools in New York. As director of the New York Institute of Musical Art, Damrosch firmly established his right to be considered among the foremost musical educators in America, even if his work in other directions had not already won him that distinction. In 1924, the Institute of Musical Art merged with the Juilliard Graduate School to form what in 1946 was renamed to The Juilliard School of Music. This institution is one of the richest of its kind in the world, and, together with other American music schools, has done much to give American students as fine musical opportunities as may be obtained anywhere. In 1898 Damrosch also succeeded his brother Walter as conductor of the Oratorio Society, which he directed until 1912. Damrosch Park, part of New York's Lincoln Center for the Performing Arts, is named in honor of the Damrosch family.