Do the sponsors of a conference have anything to do with the quality of that conference? In particular, can someone conclude something of a conference's quality by an NSF sponsorship? I searched around looking for NSF funding criteria but could find very little.
I'd suggest that prestigious organizations, including companies, will sponsor a conference only if they have high confidence in the organization and in its organizational committee. Associating themselves with less than fully reputable conferences has negative value for their reputation.
To get NSF (or IBM...) sponsorship, someone has to ask for it. The reputation of those who ask is, I assume, one element in any decision, along with an analysis of the likelihood of positive contributions of conference presenters, papers, etc.
However, I doubt that very many people use the list of sponsors to decide whether to submit to a conference or not. But, they probably use a similar quality index to that used by potential sponsors.
And, a collection of weird sponsors, might, on the other hand, indicate something negative about the conference.
This is probably field dependent. For example in computer science, conferences are typically selective and there is a hierarchy of prestige. This answer will cover mathematics, where journal publications are what similarly "counts" on your CV.
As Buffy said, funding from NSF (or any other agency) is a sign of confidence in the organizers and their plans. In practice, it also means that you might be able to have your travel to the conference funded.
But keep in mind that "quality" may mean different things. I am an organizer of an NSF-sponsored series of regional conferences, and a major stated aim of the conference is to give local researchers an opportunity to speak and interact with the community. This includes early-career graduate students, as well as faculty at teaching-oriented schools who may not have many other opportunities to present their work. We aren't selective; we give an opportunity to speak to more or less anyone who requests one.
We indeed take pride in running quality conferences -- but it's worth keeping in mind that "quality" is not necessarily the same thing as prestige.
Being "NSF-sponsored" just means that the NSF contributed some funds towards the conference.
The NSF has a limited budget and wants to give money only to conferences that are worthwhile for some purpose. (Note that sometimes the purpose is related to "broader impacts" rather than "intellectual merit". For example, one of the selling points of the conference I'm writing a grant for is that it (and the area it covers) engages professors at universities where research is not a priority.) However, a small conference grant of say $10K is a very small amount for the NSF and they're not going to expend too much effort vetting the conference.
I can think of two mechanisms by which a conference might be considered to be "NSF sponsored".
One would be an NSF conference grant, where somebody organizes a conference and specifically applies for money to support that conference.
A second mechanism is if there is some larger NSF grant, and part of that grant has been specified to support a conference. For example, if somebody writes a grant to develop educational techniques, and part of their plan is to disseminate the findings, and in the grant they propose the best way to do that is to hold a conference, and funds are requested to support that conference.
I wouldn't think that NSF sponsorship would influence a person's decision to attend a conference, unless of course that sponsorship pays the attendees' way! It's possible that the speakers may be a little bit better than those in an unfunded conference, but not necessarily. Also, means of funding has zip to do with whether the conference is appropriate for a given individual to attend.