B. W. Jones was born in Redwood Falls, Minnesota. He received his BA in 1923 from Grinnell College, his MA in 1924 from Harvard University, and his PhD in mathematics in 1928 from the University of Chicago under L. E. Dickson. Jones was a mathematics professor at Cornell University from 1930 to 1948. At Cornell, he supervised the doctoral dissertations of four mathematicians: Irma Moses Reiner (1946), Irving Reiner (1947), Mary Dolciani (1947), and William J. LeVeque (1948). Jones was a member of the University of Colorado Boulder faculty from 1948 to his retirement in 1971; he was chair of the mathematics department from 1949 to 1963.
Jones was honored by being selected to write the Carus Monograph Number 10, entitled "The Arithmetic Theory of Quadratic Forms." He received the Mathematical Association of America (MAA) Distinguished Service Award in 1971. In 1991 the Rocky Mountain section of the MAA honored the memory of B. W. Jones by naming their yearly Distinguished Teaching Award after him. He died in Wellesley, Massachusetts.