Is there a general rule as to whether to use the accusative or the possessive between a verb and ING? Like: "Would you mind me/my sitting here?", "She saw me taking the pen" and "I don't enjoy your smoking here". Thanks
When a verb takes a gerund as a verb complement --as do verbs like avoid, finish, mind and resent)--, if the subject of this complement is different than the subject of the main clause --I don't mind (Bob waiting | Bob's waiting) here--, because the gerund is considered to be a noun-like form, to modify it one should use an adjective form (his, Bob's, etc.) rather than a pronoun form (him, Bob, etc.). At any rate, this is the rationale for the possessive's use here.
However, I can't think of any cases in which the use of the possessive would be obligatory, so you're probably better off using the accusative.