I'm of a real mixed mind on this ... but not for the reasons one might think.
For years, my mother worked as the head bookkeeper in a library system. As such, she was exposed to the dilemma that librarians face over the issue.
The problem is ... well, being caught between the anarchists and the absentee parents, who both abuse what a "library" is supposed to be.
The anarchists would use the library public computers to view porn. And not artistic porn either --- the most deliberately vile, disgusting things the Internet has to offer. They weren't interested in viewing the porn themselves, but instead were watching the reactions of the innocent patrons walking by and seeing those images. They'd also print many of those images --- not because they wanted printouts, but so that the patrons printing resumes or homework assignments or whatever would have to see them as they looked for their own printouts.
The absentee parents would ... well, drop off their kids at the library for hours every afternoon while they did other things. The kids, with no particular task at hand, would wander the library, and discover items clearly not intended for underage viewing.
Of course, both sides would claim that they were entitled as "taxpayers" to that sort of activity.
The "real" answer, of course, is for librarians to intervene --- kicking out patrons who are deliberately obscene, and refusing to admit underage patrons without supervision. But that turns librarians into security guards, which is a role they certainly don't want.
I wish there was an easy answer. Unfortunately, the real answer is for everyone --- collectively --- to grow up and act maturely. Alas, 'tis often not to be.
The problem is that a library is a public space. What one user does on their computer affects those around them. In particular, I don't want my kids exposed to that stuff when we visit the library together.
It's true that some users only have access to the Internet through the public library -- but the best solution to that is to find ways to give them cheap or free access at home.
It's also true that we don't want librarians determining what we can view or see in the privacy of our own homes, but public use of a public terminal is a different matter.
These don't seem to actually be arguments against any of my points. In particular, that there are various reasons that a library patron might be looking at something that someone might consider pornographic, and that different people consider different things to be pornographic. It's awfully glib to just say the patrons ought to have internet at home, because in fact, many of them don't; they may not even have a home. Public libraries are for everyone to use; I'm absolutely unwilling to say that public libraries are for everyone to use but only to look up things that Robert Marmorstein finds acceptable for his children to see. Or, worse, the intersection of what any US adult finds acceptable for any of their children to see.
Boiling the proposed rule down to its simplest form, it would be "Don't be a dick." This seems like a clear and obvious rule, but sadly since people generally can't agree on just what constitutes being a dick in the first place, it's not a rule that can be made.
Thus, yes, I agree with your position. The best a library can possibly be expected to do is put up a sign stating "This facility caters to The Public. This includes people whose views disagree with yours, your parents', and your religion's idea of What is Proper. This is a Public Facility. We are not responsible for your sense of offendedness. You are warned."