Bollocks. (That's a technical term.) The semicolon is the correct punctuation for a particular kind of sentence structure. So on the face of it, if you want to outlaw something, it should be that sentence structure, not the punctuation that is necessary to it.
But this is one of those rules like kill all the adverbs. Many writers today do not give sufficient attention to the quality of their prose. They are all about getting their plot down on paper and are negligent of their prose and its effects. One of the most obvious ways this shows up is as the sloppy use of adverbs. But merely cutting out all the adverbs won't make your prose better. And substituting exotic verbs for simple verbs, as some anti-adverb crusaders recommend, won't make it better either. In fact, it will make it more pretentious. If you train yourself to be a good prose stylist, if you pay attention to the quality of your prose as you write, you will use adverbs appropriately where they are needed.
Similarly, people who don't pay attention to their prose often end up writing convoluted sentences that are hard to read and then try to fix them with punctuation. If you get down to semicolons in this attempt to punctuate a broken sentence it is a good sign your sentence needs to be rewritten. In fact, I would state it as an axiom that if you have any question about how to punctuate a sentence, you should probably rewrite the sentence, and, quite possibly, the entire paragraph. But really great sentences can sometimes require semicolons and taking out all the semicolons will make the great sentences worse but it won't make the bad sentences better.
Any rule that make an enemy of any part of speech or any punctuation mark is bollocks. Great prose stylist use them all. To determine if an adverb or a semicolon is appropriate in a particular sentence, you have to look at the overall quality of the prose. Adverbs and semicolons may be relatively rare is good prose, but they have their place and are something essential to felicity of expression.