Depends on the Bond. The Connery, Dalton and Craig Bonds would kick Bourne's ass. But Bourne would kick the crap out of the Lazenby, Moore and Brosnan Bonds.
Now, if you're going by the literary versions of both characters, Bourne would win hands-down. The Bourne of the novels was very similar to the Bourne of the films - a supremely skilled and trained martial arts fighter. Fleming's Bond, meanwhile, while certainly a tough cookie, got his ass kicked quite a bit in the novels. And Fleming, for the most part, never gave him any of the magical gadets or tech of the Moore/Brosnan era to save the day.