Do generous donations to U.S. universities influence STEM PhD admissions committees?

My guess is that this probably happened at least once, but that it is rare, and for it to have a non-negligible effect on admissions is very rare. Let me give a few reasons:

1) Admission for an undergraduate program at a US university is a very different process from an admission for a graduate program at the same university.

In the former case, most of the people handling admissions are admissions officers: these are staff members of the university that are in direct contact with the administration. They also meet students and parents of students in large quantities. They have a variety of motivations for admitting students, and they admit such a large number of students that any one admission decision could be chalked up to breadth / desire for (e.g. geographic) diversity / individual taste.

In the latter case, all of the people who are materially involved are faculty (usually tenure track faculty) in the department. As a tenured faculty member in a research university, my communications with students' parents outside of a quick handshake at graduation day (if I attend, which I most often do not) are exceedingly rare. When we admit a student into a graduate program, it's because we think that they are (i) likely to successfully complete the program and (ii) likely to be more successful than other students whom we did not admit. One wrongly admitted student can really drain a lot of departmental resources.

In some STEM fields, admissions are not even done by a committee; they are done by the PI who wants the student to work with them. Whether this situation is more or less susceptible to generous family donations, I can't say.

2) The idea of a "legacy student" makes less sense at the graduate level. In most cases that I know about, large donations to universities are made by alumni of that university. Moreover, a successful alumnus of university X often wants their children to go to university X, spends a lot of time telling their children that when they are growing up, is a generous donor to university X over a period of many years, and so forth. It is less likely that a child of someone who attended university X is going to want to attend a graduate program in university X: the best graduate programs in a given academic field are only mildly well correlated with the best undergraduate programs.

3) While it is very surprising to many, there really is infrastructure in place for "legacy admits" to undergraduate programs at many universities. There is no such infrastructure in place in graduate programs, so far as I know (certainly not in my STEM US graduate program). Most faculty are by nature highly independent and not very amenable to being influenced by administrators. In fact many faculty are quick to spot the possibility of influence and push back against it. If a dean in my university came into my office and told me (I am the graduate coordinator of my department) that a certain Mr. Y was the son of an extremely generous donor and that I should take that into account accordingly...well, suffice it to say that I would have to steel myself not to allow that to have a negative effect on Mr. Y's admissions.

However:

4) At the graduate level, the decision is not just admit / reject, it is admit with funding / admit without funding / reject. Most departments do have lower standards to admit students without funding. If someone came to us and said "We'd like you to admit this student to the program. His funding will be totally covered for as long as he's here. Moreover, we will give you [enough money to be able to fund another student, say]" then...well, this has certainly never happened. But would we give it some consideration? Probably we would, if the student had a broadly suitable background in absolute terms. In the field of mathematics in the US at least, being a graduate student and paying your own way is not a position of prestige.

I would guess that the answer is typically no. The administrators themselves certainly aren't looking through the list of Ph.D. applicants, and I doubt that they have any system for staff flagging such applications.

That said, they do schmooze a lot with donors (actually department chairs also schmooze more with donors than I realized), and if a donor had a child/relatively applying for a Ph.D. at the school, it would be totally natural for them to mention that to the chair of the department as an interesting fact. It would also be totally natural for the chair to mention that to the committee, though some probably wouldn't, precisely to avoid any appearance of pressure on the committee. I suspect that application would get a more careful look due to just curiosity. I doubt there would be any conscious attempt to tilt the scales, but simple attention might give them a small boost, but no more than if the parent was a friend of someone in the department, a well-known researcher in the field, or a famous person who hadn't made a donation. There are a lot of unfairnesses in the system based on how good the networks of your LOR writers are; I think that's a much bigger issue.

Since the admissions decisions are made by faculty, most of whom have tenure, rather than professional admissions staff, I think there's much less possibility for this sort of corruption; probably the indirect effect of donations getting students into prestigious undergrad institutions, and thus having an enhanced application is bigger.