Panhandling has been determined to be an exercise of free speech...in other words, it is constitutional to ask for money.
However, Springfield and Medford have ordinances in place that ban transfer of funds or other goods from a vehicle to a person in an intersection or other traffic area as a traffic safety issue (I'm not sure of the exact wording), which have been found to meet the legal standards.
If Eugene city council could bring itself to adopt a similar measure, that would solve part of the problem and make driving safer...depriving panhandlers of all those begging venues might also force some of them to leave town.