The German system of higher education was entirely state-run, so the Nazis had in principle a lot of control. This is pretty evident in things like the Law for the Restoration of the Civil Service (1933), where they fired all non-Aryan professors. Professors were expected to routinely pledge allegiance to the regime — even people who tried to stay "neutral" were required to praise Hitler in the standard fashion, making many of them feel deeply compromised in the process. (Max von Laue, a physicist, supposedly avoiding making the salute for years by always walking around carrying a large box, using it as an excuse. I suspect this is a bit apocryphal — surely it would have been noticed — but it's a nice story.)

They occasionally dabbled into actual university curricula, e.g. in the areas of science where they had very strong stakes ideologically, like biology. They could occasionally be mobilized to "politicize" other fields, like medicine and to a lesser extent physics, but this was more a matter of picking who would get promoted (or get hired for open jobs) than, say, purging people or dictating curriculum. Note that the "mobilization" in question often came from within — pro-Nazi professors using the Nazis as an excuse to purge people who they didn't like or disagreed with (as was the case with physics). Promoting people who are pro-Nazis could set the curriculum of a school but it is not exactly the same as controlling every bit of it in a top-down fashion. One should keep in mind that there were many pro-Nazis professors who were willing to take advantage of the political climate for personal gain.

They also encouraged "student activism" on a very strong scale and this could impose a lot of difficulties for professors. University youth were generally strong supporters of the Nazis from early on, and would take it upon themselves to protest, boycott, or harass professors who they thought were being insufficiently supportive of the regime or its ideology. Student unions were often the groups that were responsible for the burning of books that are so famously associated with the Nazis.

There are also a few cases of universities in other countries that the Nazis took over, and "Nazified" in a more heavy-handed way. Some of these, like the university in Poznan, Poland, were used to continue racial hygiene research programs through dissection of Jewish corpses, and had SS officers appointed as professors.

Some readings: The case of medicine is discussions at length in Robert Proctor's Racial Hygiene: Medicine Under the Nazis (Harvard University Press, 1988). Physics is discussion at length in Mark Walker, Nazi Science: Myth, Truth, and the German Atomic Bomb (Perseus Publishing, 1995), and Alan Beyerachen, Scientists Under Hitler: Politics and the Physics Community in the Third Reich (Yale University Press, 1977). The case of philosophy, and specifically of Martin Heidegger (a majorly important philosopher who had fascist leanings and was rewarded for these under the Nazis), has been gone over in some detail in Victor Farias, Heidegger and Nazism (Temple University Press, 1988). I don't know if there is any all-in-one study of this but my above answer is a mixture of these three fields as exemplars.