Boo Radley is white, and there are several context clues that tell us his race. First of all, the Radleys lived just down the street from Atticus, Jem, and Scout. During this time of the 1930's, a black family would not be living in the same neighborhood as whites. Segregation still existed, and the separation of the races would definitely be standard practice. Another clue was when Harper Lee described the Radleys and their house. She said that "negroes" whistled when they passed the Radley house. This indicates that they were scared of the Radleys and were especially uncomfortable walking by the house because of the folklore and superstitions that surrounded Boo. In addition, Boo placed a spelling medal in the knothole of the tree for Jem and Scout. This showed that Boo went to school, something that wasn't necessarily given to black children at the time. Many black children had to work to help support their families, and many blacks were illiterate at this time.
The Radleys were a well-established family in town, and that is one reason why they hid Boo away for years. The embarrassment that he caused by being arrested as a teenager was devastating to the family's reputation.