Would you mind mine smoking here?还是 Would you mind my smoking here??

后者 smoking是动名词，所以前面用形容词性物主代词修饰。

Would you mind my smoking here
my 是形容词性的
mine是名词性的
名词前要用形容词所以用my

后者，smoking的逻辑主语是I，我们需要一个形容词性的来修饰动名词smoking

Would you mind my smoking here right

Would you mind my smoking here?
介词后跟宾格

后一个正确，因为：mine是名词性物主代词，只能单独出现，在句中做主语和表语；my是形容词性物主代词，后面一定要接名词，不可以单独出现，只能做定语。