Firstly; it is true that many right wing persons and groups mistrust science for fallacious reasons.
However this is not interesting. And it does not explain the decline in trust in science. Take the theory of evolution. Many religious conservatives do not believe it. However they were certainly aware of the concept fifty years ago. They didn't believe it then either. Logically their trust level on that point can't have changed much. So; it is something else.
My opinion is that the right wing rely on their intuitions. This will sound like I'm castigating them as unscientific, but this is not true. Intuition is a genuine information discovery system, sometimes flawed, but, and this is the key point, usually right in a way that increases the odds of survival or else it could not exist.
So the question we need to ask is: why would the right wing feel queasy about science today?
Consider the following:
1. Their tribal enemy has overwhelming majorities in large scientific fields.
That by itself is enough of an explanation. The intuition that your native enemy is likely out to get you by means fair or foul is a pretty good intuition.
2. Universities teach liberalism. It is so pervasive that commonly students and professors don't realize this. They are the proverbial fish in water. If you believe conservatives have low IQs or are intrinsically biased against the scientific method, you've been drinking the kool-aid.
If universities don't teach liberalism then it is quite remarkable how synchronized the students are in their beliefs on liberal talking points. This isn't an accident and it has nothing to do with rigorous logical arguments.
If as a student you read enough old books you'll run into trouble pretty quickly because the right wing weren't the straw men identified on television shows such as the Colbert Report or The Daily Show. Quite simply there exists a strong intellectual heritage on the right and most students never encounter it because the media and faculty are usually incapable of steelmanning their opponents arguments. That does tend to happen when point (1) is true.
3. There have been no major scientific discoveries seen as directly beneficial to the human race for some time now.
This will be my controversial point I think. It is a simple one. Every field has a need to justify its existence, regardless of whether it's astrology or astronomy. Astronomy needs to provide useful data, new evidence of discovery, and astrology needs to make people feel entertained. If either field stops accomplishing those things they become stagnant and eventually they get forgotten.
This will be a bitter pill to swallow but consider that the discovery of DNA and the Human Genome Project has had no meaningful effect on people's lives. They aren't living longer or better because of these projects. Many promises have been made with each development, but no meaningful change has occurred.
Naturally there are many worthwhile scientific projects which don't directly affect us and especially not right away. However the intuition when hearing that 'something doesn't affect you but could in the distant future and also by the way could you please give us money today' is that you may be listening to a description of a scam.
Perhaps technology has stopped integrating scientific knowledge into useful products and services in the way it used to. What matters is that people are broadly looking over their lives, hearing how batteries/cancer will be solved soon by some whiz-bang-discovery and then later hearing nothing and thinking they've been hearing a lot of squealing but not much pork.
Common people use common sense, and that's saying to them we haven't been back to the moon in half a century. That major diseases remain unsolved problems. That their wages haven't gone up in decades. Eventually youthful optimism becomes lost for what is a completely rational intuition.
These are all areas science is supposed to indirectly relate to, and people aren't seeing it anymore. All the time they have heard how wonderful 'progress' and 'science' is, but not seen it for themselves outside of, say, computing or information science.
So those are my 3 reasons why a right wing person would grow to distrust science and scientists.