I, myself, do not generally give money to people who ask for it on the street (I'm not saying I never have, just that I almost never do), but it seems to me that if/when cash goes away, that could be a bit of an issue. Don't search on panhandling and flattr; it goes to a Bad Place. There are memes out there about making a homeless person smile by giving them shoes, or a t-shirt, a sandwich, a bottle of water, etc., and judging (har de har har) by some of the reporting on How Do People Who Ask for Money on the Street Then Spend the Money, I suppose some of the problem could be addressed by just offering people beer and/or cigarettes and cutting out the cash-in-the-middle (<-- I am not advocating this!)
Taking this a bit more seriously, busking and other requests for money on the street are essentially informal person-to-person transfers, which can be handled in a lot of ways: a busker probably doesn't want to stop to use a Square Reader, but you could _almost_ imagine some sort of scan-a-code system that could be implemented by Real Change (maybe a pin with a QR code on it: you scan it, donate a dollar or few using PayPal or other payments system, and the vendor gives you you the paper. Altho let's think about this for a minute. It's still a news_paper_).
I suppose this would be also an opportunity to think about what happens to bottle and can deposits, but that's actually surprisingly cash free anyway. You pay for the groceries, more often than not with a payment card, then bring the empties to a redemption center in exchange for a slip of paper. They currently cash you out (not that I've ever seen this happen, since everyone seems to be like me and buy way more than they redeem), but could as easily transfer that balance to a reloadable card.
Bottle deposits really ought to be repriced anyway. 5c is ridiculous.
The answer to the post-cash begging question came up over on Freakonomics a couple years ago and they did not give a good answer.
As we have been discussing this, R. got to wondering why we didn't see any panhandling in A'dam; they've banned it, as has Rotterdam and Den Haag either just as or is about it, I forget which.
Humor about the possibilities: http://www.youtube.com/watch?v=PxpUjAuH5U4
I thought that people who might give a $10 or might give leftovers from a restaurant meal might be willing to hand out a prepaid card to McDonald's, or even a general purpose prepaid Visa. Someone did this in the interest of Science, or, er, Journalism:
A student panhandled and wrote a book about it; people gave him gift cards, among other things: