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Discounting the actually quite good idea of having the asteroids be the site of colonization would it be better to use human miners or automated ones?
Both would probably deliver the goods by unmanned rockets. Phosphate to the farms on Venus and Mars. Transition metals to everyone. I think simple assaying, mining, and refining A.I. are completely feasible. As are packaging and shipping.
So what if any human presence is needed in the belt and why?
1. NASA mindset is to kill as few humans as possible
Space is dangerous and evil1. Even tiniest error can effectively kill you. Most of the space program is done by NASA and the west. They have the same mindset: Kill as few as possible.
It may sound obvious, but during the cold war the Russians had completely different mindset: Get the job done, and if you die in the process, we will celebrate you as a hero.
Yes, I know I am exaggerating here, but the point is that as soon as someone is killed in space, people always ask the question: Is space exploration worth it?
2. No one cares about a dead machine
If an unmanned mission fails, the info about it barely makes it to first page of news. Mission success? It will be on first pages if it is something really interesting. Otherwise, no one cares. This is extra plus for robotic missions.
3. Caution: Living in space may cause health issues
Consult with your doctor before going to space. Truth is, that we really do not know what the long-term effects of being in low gravity are on a human body. We are trying to figure out the effects on human body, and we are going to find out pretty soon.
But one we know for sure: Being in space and mining asteroids will definitely have some effects on human body. We know that it will have also some effects on machines. But we do not care about machines (see point 1 and 2).
In nutshell: We are going to use as many robots as possible.
I think it is plausible to assume that some humans will be involved in asteroid-mining operations, but the people will be dying. So there will be a moral push to replace them with robots and machines.
1: It does not mean that space it intentionally evil towards anyone. The pure nature of space is that not many live organisms can survive in space without any protection2.
2: And even when being protected, survival in space is hard. [citation needed]
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!"
Automatize as much as possible.
How much difference these factors make can be easily seen when you consider that it took a 300 ton rocket to get a 100kg robot to the moon, but a 3000 ton rocket for two humans of average mass. This simple economic reason is why no human has left Earth orbit for over 40 years.
However, there might be a few things which are hard to automatize. When robots break down, there are limits to what you can do remotely. Remote technology controlled from Earth can perform some maintenance and repair tasks, but sometimes you just need actual hands-on treatment by a human technician. Sure, when you have enough robots you can calculate in some losses due to irreparable defects, but depending on how reliable your robots turn out to be, sending a few humans in a pressurized habitat with them who repair any broken mining robots might be economically justifiable.
If its worth enough for someone to send a ship full of high tech equipment then someone might try and rob the the mined minerals or the high tech equipment.
Since that person might not have sufficient funds for his own little army of space pirate robots he will need to hire humans for the job.
Now you might be able to have a robot security force on the asteroid but human creativity or level of insanity might be something too complex and hard for an A.I. to comprehend.
Therefore you will need a human security force to guard against space pirates.
Identify yourself! Code: 1402 Incorrect, ratatata Telecommunication-Identification is of course more complex (a lot of crazy math), but still easily deployable (compared to an off-planet mining facility). And no matter who the approaching party is (human or robot), identifying wrongly results in termination. If there is a human guarding, he might be tricked, be a pirate himself,... My method works because there is no way to sneak in space. You're there or you're not. No rock to hide behind, no tree,...
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Beside from digging rocks, an effective miner has to perform geological surveys, send probes, analyze results, choose the proper mining spots and techniques. The same with refining - you must design and control the cycle depends on materials you got. That tasks could be harder for AI than just sending bore machines in direction some kind of "ore detector" said and then put all the mined materials into some kind of all-round "ore refinery". Can't see any reasons that could not be controlled remotely though.
Remember Philae? If you are seriously going to do heavy engineering, send humans. The lightspeed communications lag will be several minutes and some tunnel bore or smelter running the wrong way that long can wreck your whole day.
You use humans in preference to robots when robot technology isn't adequate. Most particularly the problem solving aspects when things don't work out as expected (and "as expected" is always a rarity in early days).
Since neither technology is adequate right now, it's not possible to answer the question without making presumptions about the state of the technology when somebody decides it must be done.
What's specifically wrong with robots right now is that they're dumb and clumsy. If a problem comes up they won't know how to fix it. Even if we upload new firmware they won't have the dexterity to fix it. We won't be able to guide them remotely because of the long round-trip times and low bandwidth for the signal.
If I had a few* billion dollars to invest in space mining my action plan would be
Find an average* size rock which can return me good money after mining. It would be economical to risk asteroid than the specially built mining tools, as we have plenty of asteroids.
Bring that rock closer to somewhere where it would be easier to mine with remote controlled robots. This can be a moon orbit or even moon surface.
