Read Anti-Intellectualism in America by Richard* Hofstadter; it's as true and relevant today, as it was back then. The decline, if any, probably has to do with the fact that there wasn't a Cold War where scientists could participate directly against a tangible threat and common purpose: “The Sputnik was more than a shock to American national vanity: it brought an immense amount of attention to bear on the consequence…
I simplify for rhetorical effect; my point is not that this is a literal description of the current state of the world but that it is far more true than it should be. Any accounting of "anti-intellectualism" that fails to take this into account and lays all the blame on "Americans" is too incomplete to formulate an action plan that will have any chance of success. It's not a one-sided problem.
If you want to fix anti-intellectualism, you first need to fix intellectualism and return it to its roots of dispassionate exploration, commitment to truth over all else and bending processes to find truth rather than bending truth to fit (politicized) processes, and return to great, foundational humility that even the press could not overplay into hubris. And they need to drop their blinders whereby they excuse away the damage that intellectuals have done while ignoring these ancient precepts and only crediting themselves their successes, because it cuts themselves off from the very object lessons that could help them return to this time-tested approach to science, which they still flatter themselves that they follow. If you fail to fix the intellectuals first, then all your effort to fix "Americans" is going to fail; you'll bend your efforts towards getting them to look at intellectuals seriously, but they'll end up coming to the same conclusions they already have about the value of intellectuals and you'll have wasted your shot.
I'm not holding my breath.