It's unlikely that a non-communist government would mine asteroids. If private industry found asteroid mining to be cost-effective, then some company would certainly find a way to mine asteroids. That suggests the answer to the question is, "whichever is cheaper". Obviously asteroid mining would have major costs regardless of whether it is done by robots or humans (e.g., delta-v for the mined materials).
Robots would be expensive in terms of development, programming, maintenance, replacement, and more subtly, possible lower effectiveness (e.g., programming might never be good enough to automatically find and extract the purest ores compared to a human). Humans would be expensive in terms of life support (air, food, water, and delta-v for the same) and "hazard" costs. Hazard costs depend on whether the pay and/or benefits would have to be good enough to attract human workers to such a risky business, or whether government regulations would require expensive and heavy safety equipment and/or minimum wages and benefits.
If asteroid mining could be made to be unskilled labor that can be performed by marginalized, exploited workers who won't demand liability and life insurance, then those workers might have a lower total cost than robots, and that would drive a company to use humans.
One likely scenario is that asteroid mining could be like a gold rush where ores are bought for a price and anyone who wants to go get them can sell for that price. It's very likely that after some company builds an infrastructure to mine asteroids, smaller organizations all the way down to private individuals would buy their own mining ships (or mining robots!) and go into business for themselves. Again, the most marginalized might only be able to afford a basic mining ship with a thinner hull (delta-v budget again) and therefore would take higher risks for the chance at striking it rich by staking a claim to an asteroid with great ores.
Note that coal mining is pretty dangerous with known, almost-certain, long-term negative health consequences, and yet coal mining is still done by humans the world over. Why? Because those humans are powerless, marginalized, desperate for work, exploitable, and cheaper than robots. At the same time, automobile assembly is much safer than coal mining, and yet robots are widely used to assemble automobiles. Again, in this case robots are cheaper - perhaps partly due to union contracts and government regulations on treatment and pay of workers, and partly because a robot-assembled auto has a more predictable level of quality and interchangability, which lowers the overall costs of making the vehicle.
One realistic best-of-both-worlds solution would be remote-controlled drones. The distances involved make it very tricky, though. You couldn't effectively control a drone in the asteroid belt from Mars (best case round-trip communication latency is about an hour). What you could do is stake out a territory in the asteroid belt, find the largest rock in the middle of that territory and put a base there. Have drone pilots stationed at that base, which would still be dangerous, but not as dangerous as human miners actually going out to asteroids and blasting and/or drilling. The drones do the most dangerous part of the work, guided by humans who can bring human intelligence to the process. This saves on programming costs, since you don't have to program the entire process, just the menial tasks. It also saves on liability since you can build one facility to keep all the human operators alive that would be much safer than an EVA suit or a mining ship. Plus you have a one-time large delta-v to get the base set up, and then your actual mining drones could be might lighter in weight so long-term fuel costs would be minimized.
Another parallel we can draw from our current economy is that if even one person manages to survive and make a large sum of money by going out alone and finding some great asteroid, others will flock to the belt - regardless of safety. The real winners there will be the companies that build and lease the mining ships - "you must sign a full waiver of liability to lease Exploitocorp's VX-3000 Super Mine Commander asteroid mining vessels". In addition, we would likely have more than a few "mining experts" selling their "Easy 12-step guide to outfitting your own mining vessel and striking it rich!"