I read through (nearly completely) at least six different threads on panhandling before posting this, so I am sure that I am writing something new. I also did not want to “hijack” the other threads by inserting my concerns there, so here goes:
God gives us many graces, and we (from the concupiscence of Adam’s Fall) misuse them. God gives us the grace of our hands, for instance, but we misuse that grace by slapping our siblings and making them cry. Our misuse of that grace, however, does not make the grace any less good or holy, because grace comes from a perfect God. Grace, more importantly, keeps coming into our lives, even when we misuse it because of our fallen state.
This is how I apply the Catholic Theology of grace to panhandling:
Every time I see a homeless person, I give money. I take the advice of Matthew 6 quite literally, doing it so that no one notices (I have an elaborate process for doing this, but it is not necessary for me to go into details).
In this case, I represent, as a Disciple (all Catholics being Disciples), the Body of Christ to the world, and have presented this man with grace (money, in this case). He might misuse that grace by buying drugs. Still, that is his problem. As the giver of grace (money), I hope that he can use the money for good reasons. God gives us grace and hopes that we use it for good reasons. We, however, can choose, through our free will, to misuse that grace. It does not make the grace any less holy. The homeless man has, through free will, misused that grace by buying bad stuff.
This does not mean, however, that I should stop giving grace. The sin of someone else does not end grace. The grace of God still continues in our lives, despite sin.
So I thought I stopped struggling with this issue by coming to this prayerful understanding a while ago. There seem to be (to me) problems with the other intended solutions posted by posters of analogous threads:
 Food option: Suppose the homeless man refuses? Do you leave? Is that not withholding grace? Grace is not meant to be withheld. Should we not keep trying with different offerings?
 Deliberately bring no cash option: Suppose there is an ATM machine every two blocks in the world? Are we withholding grace in that situation too?
I am NOT saying that people who do things like  and  are wrong or sinful or whatnot. It is just that, for me (a person of the lovey-dovey-never-angry-cutchie-cutchie-coo personality type), I would feel more comfortable in watching my grace misused (at least I gave the grace) than not giving grace at all.
From a theological standpoint, is it not the most sensible idea to just give the homeless whatever they want (cash, virtually 100% of the time) and then pray that they use the stuff wisely?
In times like this, I catch myself hoping that the church makes some infallible pronouncement on panhandling issues. Not that it ever would, LOL.