It really depends on the verb. With the verb ‘to mind’ you can use either the possessive gerund form or the participle form. It has the same basic meaning regardless of which form you choose, but in usage they are not exactly the same.
1) If you don't mind my asking.
The possessive pronoun can modify a gerund and this formal construction is preferred with the verb “mind”, because it is a polite request.
I hope you don't mind my asking……correct
2) I hope you don't mind me asking.
A present participle can modify an object pronoun and it is also considered correct. There is nothing wrong with this form, but it is less formal.
I hope you don't mind me asking……correct
The verb “to see” can also take both forms, but the possessive gerund form sounds strangely formal.
I saw him walking down the street…..correct
I saw his walking down the street…….correct but odd, sounds like something from an 18th century novel.
With some verbs “to keep” for instance, the possessive gerund form and the participle form have significantly different meaning.
He kept me talking for hours……He made sure that I talked for hours.
He kept my talking for hours…….He kept a tape of my recorded voice for hours (and then threw it away?)