Do the "humanities" serve any purpose at all?

Yes they do.
But, if taught and learnt with a certain approach. Certainly not the way we torture students with the humanities now. If I were an undergrad, I would hate taking anything in the humanities! Why? Because, the focus has shifted from helping students comprehend the world and understanding their own individuality and individual place in this world, to some horribly rotten dumbed down version of doctoral topics so that professors can then pretend to be ultra-smart in the eyes of students. No wonder then students are left wondering why there is a focus on some strange abstraction and theory when all they want is to "get the gen.ed. out of the way." So, that is what we have at the end of the undergrad experience: that the humanities are to most students something to endure if they want to get a college degree.

That is not how it ought to be. I found so much of valuable insights into what it means to be human from Tolstoy, Dostoevsky, Pushkin, Dickens, Narayan, Saroyan, Maugham, ..... I am so glad I read them and reflected on them on my own time and not in a choking classroom environment. The Grand Inquisitor by itself was a profound experience. I am willing to bet that most humanities departments rarely guide students towards the Grand Inquisitor.

The trigger for these thoughts came while reading this essay in Wilson Quarterly. In it, I liked this part the best (I don't agreee with everything in that essay):
[Many] of those who speak for the humanities, especially within the organized scholarly disciplines (history, English, and the like), have not quite acknowledged the nature of the problem. The humanities reached unprecedented heights of prestige and funding in the post–World War II era. But their advocates can only dream of such status today. Now the humanities have become the Ottoman Empire of the academy, a sprawling, incoherent, and steadily declining congeries of disparate communities, each formed around one or another credal principle of ideology and identity, and each with its own complement of local sultans, khedives, and potentates. And the empire steadily erodes, as colleges and universities eliminate such core humanities departments as classics (or, at the University of Southern California, German), and enrollment figures for humanities courses continue to fall or stagnate.