Which of these do you think is correct?
1. If you don't mind my asking . . .
2. If you don't mind me asking . . .
If you guessed 1, you're correct. The word asking is a gerund (the -ing verbal used as a noun), so it needs a possessive marker on the pronoun. If you don't mind me asking is equivalent to If you don't mind me question. This constructin is also known as the fused participle.
1. I object to his moving to New York.
2. I object to him moving to New York.
It's not him that you object to; it's the moving to New York that you object to. Here's an example that's ambiguous:
Harold did not approve of his daughter living in New York or of his grandchildren playing in the street.
Did Harold disapprove of his daughter who lives in New York and his grandchildren who play in the street? If so, then that sentence is punctuated correctly. But if Harold disapproves of the acts of living in New York and playing in the street, then that sentence should be punctuated this way:
Harold did not approve of his daughter's living in New York or of his grandchildren's playing in the street.
When determining whether a gerund needs a possessive, evaluate the sentence to determine which word is the object. Is the gerund the object, or is the noun or pronoun before the gerund the object. If the object is the gerund, the noun or pronoun before the gerund needs the possessive marker.
But keep in mind that the possessive case is not always feasible before a gerund. When the subject of the verbal -ing is heavily modified, compound, abstract, or incapable of showing possession, then the possessive case is impossible. Consider these:
He objects to the man next door putting up a fence.
What do you think about Russia and its satellites refusing to participate in the Olympics?
She disapproved of pragmatism dominating all our decisions.
Sometimes a sentence can be recast so that it's both grammatical and tidy (but watch for subtle changes in tone or connotation; I think I've change the nuance slightly in the first example):
He does not want the man next door to put up a fence.
She thinks that pragmatism should not dominate all our decisions.
Although it's not tidy, the only alternative I can think of for What do you think about Russia and its satellites refusing to participate in the Olympics? is What do you think about Russia and its satellites' refusal to participate in the Olympics? It's grammatically correct because Russia and its satellites collectively "own" the refusal, but it's still not very eloquent. I'm open to suggestions on that one.
Keep this in mind when deciding whether a possessive gerund is required: Although experts disagree about some aspects of the possessive gerund, they all recommend using the possessive before a clear-cut gerund whenever the choice has no undesirable side effects. With few exceptions, a one-word subject of a gerund-especially a personal pronoun or a proper noun-belongs in the possessive case.