Yes! Including "the" is inconsistent with the previous sentence and weakens the irony.
The previous sentence explained that people who compiled mathematical tables were called "computers". That is enough to establish that choices of articles can now suit this new (actually, very old) sense of "computer". So now, "computers" (with no article) refers in general to people who compile mathematical tables. "The computers" refers to a specific group of such people—but no such specific group is referred to. However, you can say "the computers" in the following sentence to limit that sentence to the actual clerks—that is, if you want to switch from general reference to specific reference. However, in the quoted paragraph, there's no need to do that.
Notice how "the" spoils the humor. If you were going to say (falsely) that our modern, electronic computers are human and therefore fallible, you'd say "Being human, computers are fallible." There's no "the", because you're talking about computers (in the modern sense) in general, not about a specific group of computers. Thus the irony in the quotation is partly spoiled. The idea is that to someone who doesn't know the old sense of "computer", the sentence would be plainly false, but the reader who does know the old sense can see why it's true and enjoy the contrast with the ignorant reader's interpretation. I almost didn't even get the joke because the "the" was stuck in there.
Congratulate yourself: you caught a subtle error involving an article!