would you ming my turning down the music还是would you mind me turning down the music

用my，因为mind后面接动名词或V-ing
你可以说would you mind turning down the music或者would you mind my turning down the music.前一句是ing形式，后一句是动名词。

my 固定句式 mind one's doing something

两者都可以。Would you mind + 名词宾格/形容词性物主代词 + doing.

例：Would you mind me/my smoking here?
你介意我在这里抽烟吗？
Would you mind me/my opening the window?
你介意我开窗户吗？

两个都对