The asteroids are potential sources of wealth and worthwhile materials; the number of people who will live in space is likely to be vanishingly small compared with those remaining on the Earth. Much of the processing will be done robotically. Waste disposal will be trivial. To bring the materials back to Earth will be the main role played by humans in space, because reentry vehicles (literally, just entry vehicles) will have to be fabricated in LEO or elsewhere in space in order to keep the costs of the space-mined materials competitive.
Moving large populations to Mars would always (or at least for the foreseeable) require lots of support from Earth; if something happens to the Earth, the humans on Mars would go extinct thereafter. The key to survival on Earth is to make sure the impact hazards are identified and dealt with in a timely way.