Amusing Ourselves to Death by Neil Postman, while not totally dealing with anti-intellectualism, devotes a good amount of space and attention to discussing anti-intellectual elements and the dumbing down of discourse.
One of the classics is the appropriately and directly titled Anti-Intellectualism in American Life by Richard Hofstadter which provides a good historical and foundational basis and which argues how much and how long it's been a strain through American life. It's very much a product of its time, though, and the view is a little limited, as one might expect from a historian from Columbia University.
Inventing the Egghead: The Battle over Brainpower in American Culture by Aaron Lecklider is kind of a partial rebuttal or challenge to Hofstadter, where Lecklider argues, at least in part, that Hofstadter and others narrowly defined what an "intellectual" is/was partly in response to the right-wing pushback of the time period (1950s and 1960s) to basically mean guys like them (white academic middle-class and up guys at or from the Ivies who were generally liberal in the left/progressive sense) and which ignored many of the other people (and peoples) that would and could be considered intellectuals and which sort of helped reinforce the argued divide between lower/working-class (and emphasis on Working) people and the Intellectuals.
The Death of Truth: Notes on Falsehood in the Age of Trump by Michiko Kakutani is a really good read in general but she specifically dives into the different elements (from all sides and parts of the political and ideological spectrum) which helped contribute to both the general atmosphere we find ourselves in but also specifically tied to the rise of Trump and the fake news/subjective reality environment, and Hannah Arendt and her ideas and arguments feature prominently and helped provide part of the frame she uses to discuss things.
Speaking of Arendt, The Origins of Totalitarianism would be another book recommendation, particularly because of how anti-intellectual fascism specifically and totalitarianism in general is (and which she goes into some detail discussing and analyzing). It's also just a (somewhat unfortunately) timeless book that goes into an analysis of what drives those kinds of movements and ideas.
The Age of American Unreason by Susan Jacoby is like a more up-to-date (but pre-Trump) version of Hofstadter's book and which helps, at least chronologically and thematically, in my mind, bridge the gap (with Lecklider) between Hofstadter and Kakutani. and which makes some of the same arguments as the latter.
While not specifically touching on anti-intellectualism, Chris Hayes's Twilight of the Elites: America After Meritocracy does discuss how so many traditionally respected and trusted institutions and groups have been undermined and despised, among which would be analysts and intellectuals and educators, etc. It's also very much of its time (2012-ish).
