If you don't mind my asking (possessive pronoun + gerund)
If you don't mind me asking (fused participle: obj. pronoun + participle)
If you don't mind *I asking (ungrammatical)
gerund and possessives (fused participle). Some people insist that when a gerund is preceded by a noun or pronoun, the noun or pronoun must be in the possessive case. Accordingly, it is correct to say I can understand his wanting to go, but incorrect to say I can understand him wanting to go. But the construction without the possessive, sometimes called the fused participle, has been used by respected writers for 300 years and is perfectly idiomatic. Moreover, there is often no way to “fix” the construction by inserting the possessive. This is often the case with common nouns. Thus you can say We have had very few instances of luggage being lost, but not … of luggage’s being lost.