Belcher was born into an artistic family. Her father, Stephen P. Belcher, worked in stained glass and her mother, Martha Wood Belcher, worked in oils and watercolor. Hilda graduated form Newark (New Jersey) High School in 1900, and then studied at the New York School of Art under William Merritt Chase, Robert Henri and Kenneth Hayes Miller, and privately with George Bellows and George Luks. She traveled to Italy in 1911, and to England, France and Italy, 1913-14.
Belcher was primarily a portrait painter and watercolorist. Her favorite subjects were Georgian Negros, the characters of Greenwich Village, and cats. A regular exhibitor in Academy annuals, she received the Academy's Julia A. Shaw Prize in 1926, and Thomas R. Proctor Prize in the winter exhibition of 1931. In 1934 she exhibited at the Marie Sterner Gallery in New York. A retrospective exhibition of her work was held at Castleton (Vermont) Teachers College in 1960.
Belcher lived in Maplewood, New Jersey and Pittsford, Vermont; she maintained a studio in New York until 1945.
