You just proved Draginol's point in his article 'Why don't liberals start more charities?' Since there are 'social programs' in place to help these people, you don't feel obligated, or even motivated (by human kindness) to personally reach out a hand. You keep on walking, assuming that it's someone else's responsibility, namely, the taxpayer-at-large.
No, you walk by knowing that they have a choice of a good life and they reject it utterly. They want to suffer on the streets, so why waste your money on them, a person who wilfully rejects everything their society offers them? If they want to be on their own, they can, and there's no good reason for you to support them with private donations.
It's not like they're entertaining you or providing some service you appreciate.
If a cancer patient was begging on the street, would you turn your back on them as well, feeling justified because there are government (and charity) programs to help them?
Sure. It might encourage them to go the press, the place they should have gone in the first place, and get their treatment subsidised properly (cos presumably they're begging on the streets because they've got something obscure that Medicare doesn't account for).
A few bucks isn't going to pay for a 500k treatment plan (cost enforced by US drug companies), although it might end up funding a trip to India to have it treated with vastly cheaper and chemically identical generic drugs, so maybe that would be a good idea. So on second thought I'd probably ask them what they're raising money for, and make a decision accordingly.