It works fine in questions without a negative polarity, as well as in answers to questions.
Do you mind if I smoke?
Yes, I mind.
True. But isn't that a sort of echo question?
You certainly can't say out of context:
"I mind you smoking! Please don't!"
Questions themselves
are negative polarity contexts. It would seem that at least in this case a very literal reply maintains the negative polarity in discourse. But that question is still needed.
In his (1980s) Grammaire linguistique de l'anglais, Henri Adamczewski tells the modal need can't be used in a plain assertion (you'd have to use it as a verb, wouldn't you?)
I was actually thinking about that earlier, but that's a more developed thought and better phrased than what I was thinking. Yes, good point!
According to him, modal need can be used in presuppotisional contexts only. Could it be the same for your mind?
Maybe!
Would this explain NPIs in general? Is the similarity just coincidental?
Note that NPIs have fairly complicated distributions. An interesting one is with "every":
Everyone who read any books was happy.
*Everyone read any books.
The same seems to apply for 'mind':
Everyone who minded the disruption left.
*Everyone minded the disruption.
