The easiest way would be: No corporations. If the owners of a business are liable for the debts of that business, then they will have much more of an incentive to behave themselves.
Granted, I'm not a libertarian, but I can't see how the concept of corporations (a government entitled class of investors with special privileges) could emerge under libertarianism.
Even despite the recent election results, I still have faith that society would not voluntarily accept an ideology that necessitates a cataclysmic drop in prosperity. Thus, a "libertarian" society would probably accept at least a bare minimum number of liberal institutions.
I also suspect that without corporations, businesses would be a lot smaller, limited to whatever the owners can personally monitor and control.
Also in cases such a pipelines and rivers, everybody can sue the person higher up and you will eventually arrive at the source.
For rivers this system has actually worked quite well. River property rights system were quite common in Britain and in the US. There is also direct comparison of rivers without such system and rivers with such system, and the property rights based once were much cleaner.
This of course changes when we are talking about dumping in the ocean. That another problem.
Also if somebody dumps huge amounts of waste on your property, or waste flows threw your river, then you have a intensive to go to court.
Also, in a well designed legal system you can do much better. Take the river example. If I don't like the waste flowing threw the river but I don't care all that much since I just use it as a cooling source, I might never bother to actually sue.
However if somebody wants to make a water sports related thing, he has a huge incentive to sue. So he sues the guy who uses the water for cooling. The cooling guy will say, he 'its not my waste, how about I give you my legal claim'. In some cases you might sell the legal claim for money.
The cooling guy improves his situation and makes a small amount of money for little effort and then does not have to be involved anymore. The water sports guy now has double the incentive to win the lawsuit, because if he wins he has two claims.
This is actually historically quite common. Most of the time you don't do much yourself, you just hand of your claims for a specialist.
You're basically giving a company who is tanking permission to do whatever the fuck they want. Companies bankrupt and go out of business all the time.
There's going to be enforcement no matter what. I'd rather be proactive than reactive.
Courts can't solve everything and they certainly can't prevent a great many things. Especially when there's such a simple way to avoid responsibility (shell corporations).
Shell companies are an artifact of how government courts assign legal liability. As courts would work differently in a libertarian society, unethical and criminal actors would almost certainly have different circumvention methods, tailored to the established rules of each different system. It's like asking how libertarians would deal with patent trolls. Patent trolls are an artifact of the existing system. They would not exist in the same form under a different system. The people who currently operate patent troll businesses would still exist, and they would undoubtedly still be opportunistic locusts, but threatening some other crop.
So how do you stop clever assholes from committing a series of offenses without meaningful consequence, in any system of civilization?
Most libertarian schools of thought are very heavily dependent on public reputation systems. A shell company with no established reputation, and no reputable person to vouch for it, would likely be entirely unable to rent or buy the boat or truck necessary to perform significant quantities of dark-night dumping in a manner that is not easily traceable.
And no matter how many fictitious name intermediaries you have, eventually a warm body has to do the dirty deed. If you can catch that guy in the act, you can hold him personally responsible unless he can provably pass the buck to someone else to share his blame.
In any case, I'm not certain how useful it is to say that libertarian theory does not solve a problem that is already not effectively addressed by existing social systems.
What if all of civilization is one giant scam?
It only takes five minutes to setup a shell company under a false name, sell your toxic waste to it for $1 (or some amount that "looks real") and just dump it all under the new company's name.
You can't really sue to recover damages in such a situation and you can't assume that a paper trail will be accurate or even kept.
You could hold the people at the tail end of the pipeline responsible but they may have been deceived as well. Also, what do you do when it's all robots?