Some ""conservative"" cities are better because "pro-business" laws allowed more units to be built so less people fall into homelessness. There are still examples like Houston though which has always had homelessness problems in spite of trying to build as hard as possible. Obviously it's not a probably with a single answer as there are people that are homeless for different reasons.
I can tell you locally we put people in jail when they defecate in the streets or shoot up in public. We also offer space at shelters for people that can follow basic rules - no drugs, no fighting, etc. In addition, our local food bank is fairly well stocked and funded, and several churches in the area have outreach programs. The end result is that I can walk downtown and never have to dodge human feces and rarely see a panhandler.
The reality is that you see downtown being clean but don't see the other people you've pushed your problems onto. And that's a consistent trend of trying to solve the homeless issue in America through things like bussing initiatives, relocating homeless camps and so forth. No conservative or progressive city has solved this problem because it's a societal issue with the US in general.