Probably not in near future, though may do a few centuries from now - but may live in orbit around Mars. That's because there's no way for humans to live on Mars without bringing all our microbes with us. And no way of knowing what they would do to the planet.
Plenty of people saying they will colonize Mars, but - our Outer Space Treaty is clear, also the rules on planetary protection - and you get people saying that the planetary protection is not an issue but nobody has explained how. Until someone can explain clearly how you can introduce a whole zoo and botanical garden of microbes to a planet without contaminating it - and how you can predict what they will do to the planet with any level of certainty, I can't see it happening myself.
Because- if anyone does do this - they are fully responsible for all the harm they cause to other countries scientific interests and any other interests in Mars. If SpaceX for instance introduce Earth microbes to Mars then it is SpaceX - and the US government who will be totally responsible for any harmful effects that result from that. When the time comes I think that will be a show stopper. Because it is becoming increasingly clear that there are several potential habitats on Mars. I expect that in the near future these will be confirmed as actually having liquid water at certain times of year or day - and I expect missions like ExoMars to find life in those habitats. And I expect that life to be interestingly different from Earth life and vulnerable to Earth microbes.
If all that is true - hard to see approval for humans to land on Mars until we understand it a lot better than we do now. Personal prediction. But not particularly eccentric view on it nowadays - before Phoenix this was a minority view - but nowadays - not so much -that there are quite likely to be habitats for life on Mars - widely recognized. That there would be life in those habiats and that it would be interestingly different from Earth life - less so. If there is no life in those habitats - then Mars is also interesting then as a planet without life but with habitats - there's a lot you can learn from a planet with habitats with no life on them - comparing them to Earth with life - to see what difference life makes to a planet. So that also would make Mars valuable in its pristine state. Also the value of Mars pristine for studies of past life. Final possibility - Robert Zubrin's view - is that there would be life in those habitats - but the life would be identical to what you'd expect in the same habitats on Earth. Personally I find that the least likely of all, hard to see how that could happen. Might be some cyanobacteria that got exchanged between the planets, perhaps halobacteria also - with common origin at least tens of millions of years, and I'd guess more likely hundreds of millions or billions of years ago - but bound to be many species that are unique to each planet.
There is absolutely no need for Mars as refuge from Earth. Mars will not be as hospitable as Earth in the next few centuries or even millennia highly unlikely even with megaengineering and everything working perfectly it's 1000 years and that's to a planet that doesn't yet have an oxygen rich atmosphere, and thinner than Earth's, and plenty to go wrong including civilizations collapsing and people just losing interest in the project, or some biological or geological show stopper issue.
Earth after an asteroid impact and global nuclear war combined would still be more hospitable than Mars on all counts - far more so - and would be the place you would need to restore and "terraform" to keep going - not Mars.