两种说法都是可以的.这是动名词的复合结构.动名词的复合结构的构成是:形容词性物主代词+doing,但是如果一个动名词的复合结构不是作主语,就可以用宾格代替所有格(也就是物主代词).
Would you mind my/me using your computer?用下你的电脑介意吗?
The father insisted on his son’s/his son going to college.爸爸坚决要求儿子上大学.
Mary’s (不可用Mary) being ill made her mother upset.玛丽病了,使她妈妈很着急.
His (不可用Him)smoking made his family angry.他抽烟使他一家人非常生气.