This, of course, is a matter of opinion. There is no correct answer as to how much society is responsible for this.
I believe that a society is responsible for making access to health care be as universal as possible. In a society as rich as ours, it does not make sense for there to be people who are not able to get basic health care. In addition, when people who can't get basic health care get badly sick, we end up having to pay more for them than we would have had to pay if they could have gotten basic health care and prevented the more serious illness.
This does not mean, however, that I believe that the government should pay for everyone to have health care. I think that people should have choices as to what amount of health care they want to pay for. That way, the market would decide what health care is provided, not the government.
So, I think the government should remove barriers that prevent people from getting health care, but people should have choices as to how much health care they want to buy.