Here's one reason why your giving money to panhandlers affects others in a direct negative way: The crew of panhandlers at Coburg Rd and MLK, which is organized into shifts, has taken to hanging out on the nearby bike path underpass. They are routinely drunk (cheap malt liquor from the nearby convenience store/gas station is what your contributions buy) and they both block the bike path and harass cyclists, particularly single women, who attempt to pass through. 

I believe this take-over of our bike paths by aggressive intoxicated homeless is a large part of the reason Eugene has uniquely lost 37% of its cyclists since 2009 while other jurisdictions are seeing increasing use of bicycles; that's a lot more cars on the road spewing poison into our air. If people would stop giving them money, they would stop hanging out there and move on to more lucrative locales.