Without even watching the video, I'm going to postulate that state-owned land is involved in some capacity, as is the State itself, whether directly (mining) or indirectly (leasing, permitting, or handing out mining rights on state land).
Private property rights can fix it.
In this case, an Argentina mountain was put up as collateral in the 1800's by the government for a loan, and when the loan wasn't repaid it went to the foreign purchasers. It doesn't really matter that they were foreign, money men from a different part of Argentina could have purchased it and mined it. Either way, it would be a case of privatizing public land.
With the mining came toxic waste that wound up in community river systems (contaminated mining water seeped into underground water reservoirs that form rivers). Cultural heritage sites were also destroyed (how would we Americans feel if someone purchase the land with the Statue of Liberty or Mt. Rushmore and decided to destroy those monuments?). The Argentine government put a stop to it by re-nationalizing the land (so says the video). Recently, foreign miners have been looking to re-take the land, but a citizen education campaign has so far put a stop to their efforts.
I suppose one could make the case that through the court system the victims could be compensated if miners started polluting their water supply. Of course, if it was a tiny village's water supply that was being infected, and they had to pit their lawyers up against a wealthy mining company's lawyers, they'd have a tough fight and lots of expenses on their hands (and that's if they were able to prove it was the mining chemical runoff that seeped underground and wound up the village's water supply. I don't really know the know the history of tobacco litigation, but I'll bet it was a tough fight that took many years to prove smoking caused lung damage and/or cancer). And even if the little guy won, in this video they discussed how some of the environmental impacts were permanent.
I believe in the free market, but it isn't the answer to everything. For instance, I discussed in another thread:
http://www.ronpaulforums.com/showthread.php?t=189034
how GM bought up the rail systems of 100 cities and systematically destroyed them in order to replace the trains with GM buses and cars (which helped turn LA into Smog City USA). What if a city or village decided that it would privatize its public water utilities, and a mining company had the foresight to purchase the rights to the water system? Maybe they do a decent job for a couple years, but then the mining begins and the water gets polluted. It's their own water so they don't care if it gets polluted so long as the mining profits vastly outweigh the water profits. As for the citizens they were serving, they could simply stop water service before the water became too polluted and before people started getting sick. Perhaps the effects of mining permanently poisoned the water - the citizens have lost their water source, the mountain has been destroyed, & the plants and wildlife that relied on the water have been destroyed.