And running away won't solve our problems. How long until our current earthly political and environmental issues are reproduced on Mars? We have "human" issues, not "earth" issues, simply moving the problem is not a solution. If we can't sustain our development in a civilized manner on a lush and hospitable planet I don't think we'll fare much better on Dune.
Not that I'm against colonizing Mars but I'd prefer if we did it out of a shared dream of a trans-planetary humanity rather than out of fear of self-destruction.
Humanity: Hold my beer....
Even if the biosphere were to go to hell, it would still have an atmosphere, and a magnetic field. Fantasists have wildly impractical ideas for fixing the former on Mars, but nothing for the latter.
Arrakis at least had an oxygen and nitrogen rich atmosphere.
If we're talking about something like a huge asteroid colliding with the earth and/or a huge volcanic episode that would trigger a mass instinction, wouldn't the pale blue dot still be more hospitable than Mars? We have access to technology the dinosaurs didn't have, our chances of survival would probably be much better overall.
Even in case of man-made devastation, such as mass pollution or a nuclear winter I think it's safe to assume that Earth would still be more habitable than Mars.
How about something even more destructive? Maybe something like a gamma ray burst? Well then Mars is probably not far away to put it out of harm's way, although maybe they would be more likely to be unaffected by it if it hit the other side of the planet due to its very thin atmosphere? I'm not sure.
Sure, we need to be better stewards of our environment, but I fail to see how our species is inherently worse then a meteor destroying a planet.
The argument could be made that Martian humans would be supremely motivated to move the environmental parameters there toward a survivable state for lightly-outfitted humans.
Contrast this with Earth, where space exploration is a four-order or fifth-order concern for most people, and long-term environmental issues are treated as abstract exercises in reducing economic externalities. This is because Earthlings have (mostly) had it easy. We're biologically evolved and culturally adapted to survive here, and thus don't have a strong intrinsic motivation (yet) to maintain the long-term health of the biosphere.
We could accidentally fire nukes, accidentally let a GM smallpox strain out, accidentally start WW3, we could do so much as it is. That is beside the point anyway.
Presumably the first people to go to Mars are going to be vetted for some level of education and skill. This will likely prevent the fear mongering, demagoguery and other forms of lying that allow some successful politicians argue against singularly true facts, at least for a little while. Likely until the mars colony is self sufficient.
Even then it seems unlikely you will get an anti-global warming lobby on Mars, because it is good for every business (presuming breathable air is government utility/service).
No, not immediately.
This isn't much different from other colonization missions in human history: humans from one place got sick of the people living there, and went somewhere (relatively) uninhabited so they could live the way they wanted without the political problems they had back home. Eventually, there were new political problems of course, but that took generations; for the initial travelers, it was a sensible move. And the political problems that came were likely different, as so much time had passed.
I personally feel that having a "fallback" in case something literally Earth shattering happens is compelling enough a reason to colonize other bodies in our solar system, Mars included.
That said, it's possible that even Mars' relatively low gravity is enough to stave off the health problems associated with prolonged microgravity/freefall, so that might be good motivation. If there's some lower bound on how high g needs to be in order for our bones to not become brittle, and Mars is above it while - say - the Moon is below it - then Mars is really the only option besides the moons of the gas/ice giants (and their respectively-giant gravity wells) or Venus (with its Earth-like gravity well and its hellscape of a surface).
So there's another scientific motivation: to measure the effects of low but existing gravity on human health.