There are a few points in your question that we should address differently.
Let's begin with the grammatical ones:
Is "You said that the mind is like the wind, but perhaps it is we who are like the wind. Knowing nothing, simply blowing through." grammatical? And what is Knowing nothing, simply blowing through? What does it modify?
- Yes, it is grammatical. The latter part may be considered a fragment, but this is a stylistic choice, in my humble opinion. (Read more about this point below.)
- Knowing nothing, simply blowing through is a participial phrase (also known as participle phrase).
- Knowing nothing, simply blowing through modifies either the wind or we. It doesn't matter much which one you pick. The natural interpretation would be the wind (i.e., the wind, knowing nothing, simply blowing through), but I believe that you are free to choose to read it as if Knowing nothing, simply blowing through is modifying we because the message is "we are like the wind".
Let's now address the stylistic ones:
Can I use an em-dash instead, like this? You said that the mind is like the wind, but perhaps it is we who are like the wind ― knowing nothing, simply blowing through.
- Of course, you can. There's nothing wrong with that.
Now, why is the fragment acceptable?
(Or even more appropriate because it creates an effect, and makes the message more powerful.)
- To address this point, I'd like to quote one of my favorite books, Artful Sentences: Syntax as Style by Edward Tufte, and let the book address this point. It's a bit long, but it's worth reading, in my humble opinion.
[...] And what was probably regarded as the greatest of all student sins, FRAG. I remember a teacher long ago who announced that any student paper containing a fragment automatically received an F, unless the student had labeled the fragment "intentional."
Nowadays writing teachers are more likely to recognize that many professional writers use sentence fragments repeatedly and artfully, often several at a time, in criticism, journalism, fiction, biography, history, essays. Often the fragments are linked semantically or syntactically to words in the preceding and following sentences. Some are like appositives in their relation to words that precede or follow. Fragments are still usually avoided, however, in legal, medical, scientific, and engineering documents, treatises, articles, and even correspondence.
On page 44 is a good example (again, a little long, but quite pleasant to read):
Although a series of fragments can create a lyrical quality, they can become tiresome. But here in a vivid memoir, ten fragmentary sentences, each an elaborated noun phrase, form four paragraphs of metaphorical, poetic expression of relationships and experiences:
Lights reflected. Lights of white-hot friction, of gases burning, of the incredible burning of ice. Lights that strike, then vanish in a flash.
Lights that strike and go on even after flashing, having set fires. Lights that blot out, lights that dwindle. Lights that cycle predictably, and lights whose cycles you cannot foretell.
Lights glowing warm on wicks, inside glass chimneys. Beckoning the snowshoe-er from the windows of a cabin collared by snow.
Lights long lost—squelched throughout a long journey, like the light of a comet—then igniting finally in the soul. Soul igniting as the blue lights reflect its image back to it, saying, The light of nature, born of the universe, is in you.
Trudy Dittmar, Fauna and Flora, Earth and Sky, 216
There are several more nice examples in the book. I hope that the above passages and the example can convince you that (nowadays) many professional writers use sentence fragments repeatedly and artfully.