Good question. Someone else pointed out that _all_ societies have benefited from destroying the environment. I get the feeling they were pointing this out to argue that "we" (modern day Americans) don't really need to feel responsible. "Everybody does it, so it must be okay."
I don't know what the answer is to the question of what actions I (and "we") should take because of my responsibility (and our shared responsibility). I do know that the responsibility I feel is greater the more tightly I am tied to a society or community. I don't feel much responsibility at all for homo sapiens' eradication of Neanderthals (assuming that's what happened). The actions happened too long ago and the "society" is too loose (homo sapiens). I do feel some responsibility for what my country's recent forebears have done; I feel a tie as an American and 100 or 200 years in the past is just a handful of generations.
For me it seems the minimum is to identify myself as an "environmentalist". Many (most?) of the objections to proper care of the environment come from people who argue against it because doing so will lessen business profits by a little bit. Even assuming it's true that it will lessen profits, seems like we need to devote more of our resources to preserving the environment. At a minimum, that should mean that businesses in our society should have to accept somewhat lesser profits (perhaps by being taxed) in exchange for better preservation of the environment.
Also, I have no problem with businesses that affect the environment negatively bearing much more of the cost to replace past harms. In the past, and even now, we've allowed businesses to pollute our environment (air, water, land) without them paying for the damage they've caused. They are among the prime offenders, although everyone does share some responsibility for being part of a society that's let those businesses operate as they have.