The Lugar Center, a nonprofit think tank founded by a Republican senator from Indiana, calculates bipartisanship scores for legislators during their time spent in elected office. Joe Biden ranks 47 out of 250 current and former senators scored for their bipartisan work (Maine's Susan Collins ranks highest among currently-serving senators, at no. 2). The index is based on a senator's track record of attracting opposition party co-sponsorship of bills they sponsor, and also how often they themselves co-sponsor bills originated by the opposing party.
Biden and his supporters often discuss his role under President Obama as the administration's bipartisan dealmaker, most notably in winning necessary Republican votes to pass an economic stimulus bill during the 2008 recession.