We see a lot of pictures of homeless people. This is usually justified as some sort of social commentary or consciousness-raising exercise. Some good background reading here, by the by, is Sontag's little book Regarding the Pain of Others which as usual isn't exactly the last word here but will get you to think of some stuff.
Most photographs don't change the world. We sort of think that they do. "Migrant Mother" and so on we think of as important photographs that were the prime movers of certain important changes. This isn't really true, and as a consequence, photographing homeless people isn't just trite and exploitive, it's (mostly) useless.
First of all, try to think of all the iconic world-changing photographs you can. Can you think of ten of them? I can think of perhaps four or five off the top of my head.
Now subtract out all those photographs that were in the hands of a skilled propagandist, that were used as a part of a larger campaign, which campaign was applied to a population that was poised to change already. Are there more than zero left?
We think of the pictures as the prime movers because they're the thing we remember. We don't really remember the speech as well. We don't remember the editorials. We don't remember the mood, and the idle conversations that encapsulated our mood and the mood of the society. We are visual creatures, and so we remember the picture.
Your photograph of the homeless guy isn't going to set the world on fire. It might, though, persuade a couple of your friends to give a few bucks to the local shelter, and that's certainly not a bad thing.
You could make the argument that (as noted above), in order for society to make profound changes it must be ready to change. Perhaps photographs of homeless people prepare society for these changes. Perhaps my photographs of poverty help with the gradual, imperceptible shifts of attitude which render the society ripe for change at some unknown future time? This is not an unreasonable hope, but it is a bit tenuous.
One could as well argue that the constant stream of photographs of the homeless and impoverished instead desensitize us. We react to these photographs in a variety of ways: Gladness that it is not us in the photo, a desire to change and improve the lot of the people in the photograph, a contrary desire to leave things exactly as they are in hopes that we will not end up like the person in the photograph, and probably many more. These reactions can, of course, overlap. Humans are fully capable of exactly contradictory and simultaneous emotional responses.
Ultimately, the ways in which societies change and evolve seems to be too subtle and complex for us to understand at this time. Arguments that our pictures aid this change, or direct these changes in a particular direction, are therefore a bit suspect at best. They are hopes and aspirations, not really arguments.
Should you photograph the homeless? I can't answer that. The question is complex and difficult, though. The answers are not obvious. Your motivations as a photographer are certainly not clear to us, the viewer, and it seems likely that they are likewise unclear to you, the photographer.