***** NOT A TEACHER *****
Hello, English:
1. I do not like your asking that question.
2. I do not like you asking that question.
According to my books, some native speakers see a difference.
To some people, #1 means something like: I do not like the fact that you asked that question. (Maybe the answer would be too embarrassing, etc.)
To some people, #2 means something like: I am happy to answer that question, but I will not answer that question because it comes from you, whom I do not like.
NOTE: As a teacher in this thread reminded us, some native speakers use the objective form ("you") when they probably want to give the meaning of the possessive form ("your").
For example, "Why does me wearing red shoes upset you so much?" instead of "Why does my wearing red shoes upset you so much?" (It's the "wearing" that upsets you so much, not the person who is wearing those red shoes.)