Robbins studied at Newton University in Baltimore. He moved to New York City after college, studied under James M. Hart in 1859, and opened his own studio in 1860. He accompanied Frederic Church to Jamaica in 1864 and continued his studies in England, Paris, and Switzerland in 1865 and 1866. He had a studio in the Adirondack Mountains of northern New York State near the studio of William Hart.
He was elected a member of the Century Association in 1863 and exhibited frequently at the National Academy of Design (1863-1894), the Pennsylvania Academy of the Fine Arts (1862-1864), the Boston Art Association, and the Brooklyn Art Association (1862-1883). In 1864 he was elected an associate of the National Academy of Design, a full member in 1878, and the recording secretary four years later. He was a member of the American Watercolor Society and the Artists Fund Society. He painted both oils and watercolors, but his specialty was watercolor landscapes.
In 1890 he studied law at Columbia and was admitted to the bar in 1902. He was a trustee of the New York School of Applied Design for Women, a fellow with the Metropolitan Museum of Art, and a member of the Century and Lambs Clubs.
His last known address was New York City. His works are on display at the Adirondack Museum, NY.