The major threat of climate change is not to things so expansive as atmospheric viability, but only to present geography and weather cycles. There will still be breathable oxygen in the atmosphere, there will still be water, animals, and vegetation on the surface. As dystopic as a post-climate-change world may look, it undoubtedly looks better than Mars.
This is probably even true after massive nuclear winters, etc. I don't think there's anything humans can do that would so disadvantage Earth as to make another planet in our solar system better suited to human life. The only thing that may require external planetary colonization would be an extinction-scale geological or astronomical event (i.e., something totally outside of human control).