I've worked with addicts for a long time. It isn't just a dollar. It's subsidizing a lifestyle that is incredibly dangerous, demeaning, dehumanizing, and debasing.
(how's that for alliteration?)
There are many ways to help people in need. If you care about the needs of the homeless, then donate to the local Salvation Army, or Goodwill (which helps people build employable skills), or Disabled Veterans of America. Or, for that matter, to the local battered women's shelter. All of these programs help people help themselves. They are designed to get people out of dangerous situations and into a life that is positive and healthy.
Being on the street is a horrible way to live, and you are literally helping people to stay there. It isn't glamorous, it isn't adventurous. It's a terrible, terrible life. These aren't rugged individualists who don't fit into the cogs of the machine, they are human beings trapped in a terrible degrading struggle, in most cases with addictions that strip them of even their humanity and dignity.
You call it over-dramatic, but I'm full up on know-nothing do-gooders who do more harm than good.
A basic rule that most of us can agree on, no matter what our background or religious preferences is: First do no harm.
That dollar does harm.