You're going into this with a lot of presuppositions and assumptions, and that makes for a poor understanding of history.

There had been growing concerns about nuclear energy prior to meltdowns. The idea that reactors could be catastrophically risky (in that the consequences of even a low-probability event were going to be very large) was understood by the late 1960s, and various groups, notably the Union of Concerned Scientists (who were not cranks), pointed out that many of the designs in operation could have things go wrong very easily, and that the regulatory infrastructure in the United States in particular was "captured" by the industrial interests.

When the Three Mile Island accident occurred, it appeared to vindicate these fears. Three Mile Island was not an issue of "poor practice," it was of a small technical mishap (the stuck PORV valve) that was one small part of an enormously complex system. It made it very hard to diagnose in real-time, and there were other aspects to the design of those reactors and their control rooms which made it very difficult to get a handle on the issue in a crisis. This, combined with the fact that there were untested and chaotic aspects to the crisis response (e.g., it wasn't clear who was in charge of the reactor during a crisis — the company that operated and owned it, the Nuclear Regulatory Commission, the state of Pennsylvania, the federal government, etc.) made it clear to many that these were not run-of-the-mill technologies. Even if you were inclined to blame it on human error, you still have to reconcile that with the fact that humans are always going to be involved with such operations, and thus human error becomes, over the long term, inevitable.

One might suggest there was a "crisis of confidence" with regard to reactors: many people did not believe that the companies which constructed and operated them, or the agencies that regulated them for safety, were actually willing and capable to do the job right. So to address your contention directly: who, exactly, would be improving on the nuclear technology? If the answer is, "the same people who made the faulty versions, and were willing to cut corners to make more money," then that is not going to be reassuring to people whose fundamental suspicion is with the whole "system," not just the reactor as a piece of technology.

It should be also noted that the fortunes of the nuclear industry in the USA were not merely affected by public opinion. They were already in a slump for financial reasons: nuclear reactors are extremely capital intensive, and require decades to make good on their investment. Even prior to Three Mile Island the orders for new nuclear plants had flat-lined; the added public perception and lawsuits were the nail in a coffin. The result is that most of the nuclear plants in the USA today are very old, which is the main reason they are being shut down: not because society has turned against them in a new way, but because they are reaching the end of their operating life anyway. The issue is that new ones have largely not been built in the intervening time, which is why the total number is declining.

Chernobyl of course resurrected the issue, even though in many ways the Chernobyl accident is specific to the Soviet context (both in terms of reactor design/operation and crisis response). But it reinforced the idea that nuclear reactors are high-risk technologies: low probability of failure, to be sure, but very high consequences possible as a result.

Does this stance have unintended consequences? Indeed. Removing nuclear power from the "mix" has tended to mean an increased reliance on fossil fuels, many of which have no chance of catastrophic accident but whose burning creates definite long-term health and environmental hazards. But many studies have shown that when it comes to weighing risk, people are more willing to accept small-but-sure risks that happen every day ("mundane" risks, like driving a car) than they are unlikely-but-catastrophic risks that involve the unfamiliar (and nuclear technology counts as that for most people, a "dread" risk). This is a psychological element, and no amount of concluding that it ought not be the case that people think that way about risk will change the fact that people do think that way about risk.

For an excellent history of Three Mile Island and its affect on te nuclear industry and public opinion, see J. Samuel Walker, Three Mile Island: A Nuclear Crisis in Historical Perspective (University of California Press, 2004). Walker is exceedingly balanced; he was the official historian for the US Nuclear Regulatory Commission. On the development of attitudes towards nuclear power and technology in general, I do love to recommend Spencer Weart, The Rise of Nuclear Fear (Harvard University Press, 2012). Weart's is a cultural/scientific history that shows how these attitudes evolved over a long period of time — Three Mile Island played upon existing issues that had been germinating around nuclear matters since the early 20th century. And lastly, if you are interested in how people judge various types of risk, the work of Paul Slovic is very useful, esp. Paul Slovic, "The Perception of Risk," Science 236 (17 April 1987), 280-285. And in thinking about risk and complex systems, the classic sociological book is Charles Perrow, Normal Accidents: Living with High-Risk Technologies (Basic Books, 1984).

The only thing I would add to restricteddata’s answer is that it’s very debatable whether nuclear power can be considered “green energy”.

In some ways it is. It’s relatively low in greenhouse gas emissions, with most of them coming in the initial construction. (The curing of concrete produces a lot of carbon dioxide.) It’s also relatively sustainable in the sense that nuclear fuel produces far more energy per ton than any fossil fuel, so there isn’t much danger of running out of it anytime soon.

But the enormous cost of cleaning up and storing the spent nuclear fuel - and the radioactive pollution that can’t be eradicated fully for thousands of years - keeps it from being as environmentally friendly as solar, wind, or geothermal power. Much of the cost-effectiveness of nuclear power was based on an underestimation of how expensive this cleanup would be.