The answer is: not really.
These two posts at Language Log explain:
Danglers aren’t grammatically wrong the way a sentence like “I bacon the ate” (as opposed to “I ate the bacon“) is wrong. Lots of native speakers of English use danglers, as do lots of native writers of English.
That said, danglers can be bad writing, which is reason enough to know what they are and how to avoid them.
Danglers aren’t always bad writing. Unfortunately, I lack the linguistic expertise to explain the difference between a good dangler and a bad one, so there the matter stands. For the moment.*
[UPDATE 1/26/2016: explanation coming shortly]
I post danglers on English 109 because I like them; they tend to jump off the page at me (most of the time).
I also post them because I’m fairly certain that looking for the dangler in a sentence helps students see syntax. “Seeing” the syntax of a sentence isn’t easy if you were never taught formal grammar or sentence diagramming.
I’m in that category myself. Before teaching English 109, I had learned most of what little I knew about grammar in Spanish class. I wrote grammatically, but I wrote “by ear.”
When I began to learn the formal categories of grammar and linguistics, I was thrilled to finally find out what the sentences I had been writing for all these years were actually made of.
*Sentence fragment intentional.