Bikes may not solve the issue of homelessness, but it seems like they can certainly improve the quality of life for those who have no homes:
This article explains how a large percentage of the transient population in Portland depends on bikes to at least get them back and forth to jobs (in a much more affordable way than public transit). Bikes also help them have their own mobile enclaves of living space.
It's not like bikes solve homelessness, but it does seem to me that they are a pretty large factor in helping people without homes or lots of money have access to services, jobs, and other necessaries that they might otherwise have a hard time accessing.
(Randomly enough, the guy with the big old white beard and cowboy hat in the pictures from the article is a guy I saw almost every single day on my way to work--him and his two buddies, always camped out by the river.)