I'm not sure you can stem the flow. And I believe there are mathematical illusions that compound the problem.
Any effort to stem the flow requires the desire on the part of the beggar/homeless/whatever. For sake of this post, beggar will do. For various reasons, mental, medical, physical, these people aren't part of society as we know it. And, these sorts of people have never really been part of society as we know it. Sure, we can mention a few who have had troubles and come out of it, but statistically, that's closer to 0 than anything.
The key problem to a resolution to the issue is that these people have rights. You can't do a thing for them if they don't want to participate in it. There have always been those who don't want to participate in society in general. Lacking other resources, homeless is where many end up. By choice, even if impaired choice.
It has always been so.
Today, there are fewer cracks for these people to disappear in. There is a greater population than ever so even if the percentage of homeless hasn't changed, the number has. And when you put a large number of people into a small crack, the problem appears bigger. Never before have we been able to see and count them as well as today. This, as much as anything else contributes to the increased number of known homeless.
So it is not so much that the problem is any bigger by percentage of population (a simplification, yes), but that the sheer numbers in that population are no longer ignorable.
Even assuming money was no problem, on what basis do you force these people to accept treatment, medication, housing or any other amenity. For even though we can't help them all, so many don't even want what is available, even if it's only pride in fending for themselves rather than taking a hand out. Or up.
Indeed, there is probably a second mathematical illusion that works against them. A single person down on his luck assesses his chances for improving his lot. Yet, that same person down on his luck with a bunch of other folks often percieves his chances as much worse even though his individual chances didn't change any at all.
While I don't believe it will ever go away, that mathematical issue should not prevent us from aiding who we can and who will accept it. And by the same token, our aid should also be voluntary, and not forced.