Which is correct?
- If you don’t mind me asking?
- If you don’t mind my asking?
Because “asking” is a gerund, which is a verb used as a noun, “my asking” is correct. In this instance, “my” is a possessive adjective modifying “asking”, the noun.
Reminder: his, her, its, my, your, and their are possessive adjectives or pronouns.
A way to get around uncertainty when writing is to rewrite a sentence. For example, “If you don’t mind, may I ask you a question?”
However, one cannot get around a grammar question on a test!