Rationality could be defeated by one powerful enemy (e.g. religion), but also by a concentrated attack of many diverse enemies. These days I would be more worried by the latter.
Rationality is a common enemy of many beliefs. If you refuse one specific truth, you must also refuse other specific truths this one is connected with, then you must refuse the general rules of reasoning, and the whole meta-level. And this is where it becomes dangerous: people with different false beliefs can be opponents at the object level, but allies at the meta-level; they may all agree that all this talking about "evidence" is bullshit... some of them because it offends their religious beliefs; others because it leads to politically unacceptable conclusions; yet others because it can be used to support sexism or racism; etc. Each of them wants to remove some specific conclusion; all of them want to stop the same algorithms for reasoning.
I don't think it's so likely that the science in the west could be destroyed these days by religion alone. But it could be destroyed by systematic attacks from all sides: religious people who hate hearing about evolution, paranoid people who hate hearing about vaccination, libertarians who hate hearing about global warming, social justice warriors who hate hearing about differences between people, even the average Joe who hates being told he is wrong about anything... all of them can together agree (and vote, democratically) that scientists should just shut up when their results are inconvenient. And if this becomes a social norm, then scientists are pretty much only allowed to agree with the public opinion, and invent some new harmless gadgets.
And the risk is that a) science gets associated/aligned with political groups and b) someone of the other policical groups gets powerful and - using methods of his opponents - uses the meta-level as a unifier to bring down the science-faction.
Paper by the Cultural Cognition Project: The culturally polarizing effect of the "anti-science trope" on vaccine risk perceptions
This is a great paper (indeed, I think many at LW would find the whole site enjoyable). I'll try to summarize it here.
Background: The pro/anti vaccine debate has been hot recently. Many pro-vaccine people often say, "The science is strong, the benefits are obvious, the risks are negligible; if you're anti-vaccine then you're anti-science".
Methods: They showed experimental subjects an article basically saying the above.
Results: When reading such an article, a large number of people did not trust vaccines more, but rather, trusted the American Academy of Pediatrics less.
My thoughts: I will strive to avoid labeling anybody as being "anti-science" or "simply or willfully ignorant of current research", etc., even when speaking of hypothetical 3rd parties on my facebook wall. This holds for evolution, global warming, vaccines, etc.
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Also included in the article: references to other research that shows that evolution and global warming debates have already polarized people into distrusting scientists, and evidence that people are not yet polarized over the vaccine issue.
If you intend to read the article yourself: I found it difficult to understand how the authors divided participants into the 4 quadrants (α, ß, etc.) I will quote my friend, who explained it for me:
I was helped by following the link to where they first introduce that model.
The people in the top left (α) worry about risks to public safety, such as global warming. The people in the bottom right (δ) worry about socially deviant behaviors, such as could be caused by the legalization of marijuana.
People in the top right (β) worry about both public safety risks and deviant behaviors, and people in the bottom left (γ) don't really worry about either.