Any extinction-level event like one we've got traces of would leave the Earth far more habitable than Mars, and keeping the species alive and thriving would require more concentration on what was happening to Earth.
Assuming Earth was eaten by a giant mutant space goat, consider what the species would need to survive on Mars. It's not possible to live on Mars without an advanced civilization and economy, and for species survival this would have to be completely supportable and expandable on Mars. My gut feeling is a minimum population of a million, with sophisticated resource extraction and manufacturing facilities. That isn't going to happen any time soon, and it isn't clear to me that waiting a few decades to send humans to Mars is going to matter.