- Do you mind my smoking?
- Would you mind shutting the door?
After would you mind/do you mind we normally use an -ing form. It is possible for the -ing form to have its own subject.
- Would you mind opening the window? (= Will you please open the window?)
- Would you mind my opening the window?(= Can I open the window?)
Note that after would you mind/do you mind we use a pronoun in the possessive case.
An if clause is possible after would you mind/do you mind.
- Would you mind if I opened the window?
- Do you mind if I smoke?
In general questions about what people think of something, we do not normally use would you mind.
- Do you mind the smell of tobacco? (General question)
- Do you mind/would you mind my smoking?
- Do you mind people smoking in your house? (General question)
We can use don’t mind/wouldn’t mind to say that we are not troubled by something.
- I don’t mind your coming late.
- I don’t mind your opening the window.
- I wouldn’t mind if you smoke.