Fordarkness tells the story of his mother selling some furniture that she was asking $500 for $25 to help out a couple who'd been going through some rough times, and he says that it seems like a good idea to help beggars out when you believe they genuinely need it. I've never begged for money or powerleveling (although I have begged same-level guildies to tank or heal an instance for me, I'll readily admit), but, like Fordarkness, I have been gifted money by friends who saw how close I was to my first mount.
Of course, guildies are a different situation-- I'll give them anything they need, because I figure you might as well show loyalty to someone besides yourself. As for beggars, though, I've never helped them, and unless they take an effort to show my why they actually need what they're asking for, I'll keep it that way. Have you ever felt the need to make a lowbie's day and gift them that mount money or run them through that instance? Or does a tell from someone you don't know asking for something automatically add them to your ignore list?
Update: Right after I finished writing this article, reader Preacherman23 sent us this elegant solution: one player puts 50g in the trade window of a beggar... and then just goes afk for an hour or so.