What is the meaning of “crew-cut Democrat”?
Every reference I can find by web search is either to this quote about Hamilton, above, or to "Ronald M. Mottl [sic] a tall, crew-cut Democrat, whose Czech name is a decided asset in the burgeoning middle-class suburb of Parma." NYT 1970
Since that's it, it's not apparently really an expression; we'll just have to look at the text to see what the authors mean -- and guess.
So we have two clues: one's a "throwback" (the Hamilton reference is from this year; certainly crew cuts are less popular than they once were) and the other's campaigning in a middle-class suburb.
I think in each case we are hearing that these Democrats had literal crew cuts (else it would be confusing). The image Woodward has above is to a time when crew cuts were popular, so it's either someone old-fashioned or someone who never bothered to update a superficial thing with the times. The NYT reference is from 1970, when longer hair was getting started. Mottl wasn't part of that new trend. This also shows him as either a little old-fashioned or at least unconcerned with trends. Both references suggest a certain (social, not political) conservatism, a nonconformity to new trends, though Mottl, in 1970, would have been in good company among his constituents.
The full statement, for a reason we do not yet understand, is not intended to be positive towards Lee Hamilton.
Crew cuts have been explained elsewhere, i.e. short, bristly, army style post World War II haircut - low maintence. But it coud be helpful to see the cut as distinct from a 'conservative' "short back and sides" type of haircut. The latter is to look smart for work, etc, the former gives a more tough or rugged look. The concept can be imagined, as an example, in the film ' Snatch' by Guy Richie where, to get a descriptor (and measure of) of an adversary he had not seen, Vinny asks "Does 'e look 'ard", i.e., does he look hard (tough, strong, violent, etc) (https://en.wikipedia.org/wiki/Snatch_(film)?
Outwardly apperance, including hair cut speaks volumes. See also Dolph Lundgren in the film Rocky IV for an exagerated crew cut (https://www.google.com/search?client=firefox-b-d&q=rocky+iv). However, these impressions can be based on feelings of prejudice (often irrelevant to the argument), rather than facts, reason, and logic (https://examples.yourdictionary.com/ad-hominem-examples.html)
The crew cut descriptor may be a well hidden or subtle ad hominem attack. This type of attack is described in Google dictionary as(of an argument or reaction) directed against a person rather than the (political) position they are maintaining. This is colloquially described as one person having a dig at someone else, i.e, making a mocking, sarcastic, or insulting remark, gibe, or criticism at someone's or something's expense. One can also look up the more modern: throwing shade (https://www.google.com/search?client=firefox-b-d&q=throwing+shades) as in "There was an awkward situation when the singer seemingly threw shade at pop legend X".
The dig, and thus the impact of the sentence, is deepened with the inclusion of the term throwback, which in addition to the explanation in answers elsewhere in this post can, in the extreme, include troglodyte, Neanderthal, etc., due to heavy set (beetle-browed) eyebrow ridges, brutish or stupid looking or, say, acting like a male chauvinist (https://www.atlasobscura.com/articles/please-stop-using-neanderthal-as-an-insult-say-neanderthal-experts,https://www.haaretz.com/archaeology/MAGAZINE-archaeologists-find-hybrid-neanderthal-denisovan-in-siberian-cave-1.6409494.
Additionally, the reference to Southern Indiana can have a double meaning (see also double entendre). Meaning 1) Innocent geolocation information relaing to the Lee Hamilton's political constituency. Meaning 2) where Native Americans and captured Africans were enslaved in America. We mostly gleaned information regarding the slaves located in Southern states of America (https://en.wikipedia.org/wiki/History_of_slavery_in_Indiana, https://en.wikipedia.org/wiki/Southern_Democrats).
Meaning 2 remains a 'thorn in the side' of the previous President Trump and his supporters often called rednecks: https://www.google.com/search?client=firefox-b-d&q=what+is+a+redneck, https://apnews.com/article/race-and-ethnicity-media-slavery-cultures-donald-trump-afdfa0b3454d5a7fdf544158ee1f5e1f) and;https://www.theguardian.com/commentisfree/2020/nov/19/us-election-showed-republican-racist-southern-strategy-falling-apart.
Furthermore, the mention of the "...served thirty-four years in Congress..." can be seen as a dig at someone who is culturally 'resistant to change' at least with regard to a job. This type of person stays in the same job forever (although this used to be the norm - see 'Cradle to Grave job') although this concept is now seeing a revival (https://medium.com/@pragatisharma3/why-we-outgrew-the-concept-of-one-cradle-to-grave-job-18ebc26f920a, and;https://www.theguardian.com/politics/2013/apr/08/what-is-thatcherism-margaret-thatcher).
Thus, the crew cut comment is the reinforcement to an array of derogatory opinions