Is Jackson Rathbone on Twilight related to Basil Rathbone?

Yes he is.

"[Basil] Rathbone married actress Ethel Marion Foreman in 1914. They had one son, Rodion Rathbone (1915-1996), who had a brief Hollywood career under the name John Rodion. The couple divorced in 1926. Rathbone was involved briefly with actress Eva Le Gallienne. In 1927, he married writer Ouida Bergère. Basil and his second wife adopted a daughter, Cynthia Rathbone (1939-1969). He is distantly related to American actor Jackson Rathbone, who starred as Jasper Hale in the Twilight Saga."