Much of the reason why anti-intellectualism is held in high regard is because of the historical basis of the intellectuals being apologists for the rulers of civilization. Whether it's Milton Friedman being friendly with Augusto Pinochet who had no qualms throwing people out of flying helicopters that didn't comply with his right wing ideology or Trofim Lysenko who had his opponents killed or exiled under Stalin, intellectuals have a storied history of propping up bad regimes and bad policies so that they can bargain with said regimes to execute their own side projects (ex. Lysenkoism). Part of that comes from the fact that they believe their pursuits are divorced from the social and political dimensions when in reality they're so heavily tied to them that it's inescapable. Like it or not, they need to realize that they must be part of the social and political dimensions to mitigate the dangers of anti-intellectualism. That means there needs to be some form of science education/advocacy (like John Dobson did in some regard with astronomy or Carl Sagan and Neil DeGrasse Tyson had done for astrophysics) and participation in the political process such that the common person (the non-intellectual/scientist/expert) can digest what is said and done by intellectuals rather than cloistering the knowledge away behind thick books and thicker language. I'm not sure any of those suggestions can yield positive results but it seems it's better than cozying up to the current US President or hiding behind some tech billionaires to pursue your dreams of Martian colonies.