I'm at Stanford, and if you think that Stanford doesn't use every bit of its institutional influence to promote liberal political power, you have rocks in your head. This university, like its peers, works tirelessly with media such as the NY Times to promote their candidates. They aren't allowed to donate institutional money directly to a candidate, which politicians need to buy media ads to get their message out. Instead, Stanford only employs those who support the institutional liberal political agenda (calling this policy "celebrating diversity"), then those people go to their partners in the media as "Stanford professor so-and-so" with "news" about what they've just "discovered" that has implications for how you should vote: "Stanford professor finds that [conservatives are mentally ill, people who vote for conservatives are bad, Republican claims are wrong, Obama is awesome, businesses need more regulation by liberals with elite degrees and no business experience...."]
Conservatives have to donate money to buy ads. Stanford gets to post its political campaign messages directly as "news."
Meanwhile back on campus, every effort is made to indoctrinate thousands of students and send them out as an army of "individuals" to do heroic things in the service of those who people at Stanford are expected to support. I had to break off my work a few weeks ago and go to another building when the second floor was taken over by a law professor who was leading a pep rally for Obamacare called a "discussion of the issues."
The notion that universities such as these just have some left-leaning individuals acting privately while the institution itself remains resolutely neutral politically "seems rather disingenuous to me."