Balfour was born in Toronto, Ontario in 1882 and was educated at the Hamilton Collegiate Institute. He attended the University of Toronto, earning an M.B. degree in 1906 and M.D. in 1914. During his internship at Hamilton General Hospital in Hamilton, Ontario, he worked with the surgeon Ingersoll Olmsted, who recommended him to William James Mayo and Charles Horace Mayo. Balfour joined the Mayo brothers' practice in Rochester, Minnesota, in 1907, initially as a pathology assistant. In 1909 he became a junior surgeon and in 1912 he was promoted to head of a division of general surgery in what is now known as the Mayo Clinic. In 1910 he married William J. Mayo's daughter, Carrie Mayo.
