The structures Would you mind…? and Do you mind…? are often used to ask people to do things, or to ask for permission. After would you mind / do you mind, we can use –ing forms or if-clauses.
Would you mind opening the window? / Do you mind opening the window? (= Please open the window.)
Would you mind moving a bit? / Do you mind moving a bit?
Would you mind my smoking? / Do you mind my smoking? (= Can I smoke?)
Would you mind if I opened the window?
Do you mind if I smoke?
We can use Do you mind…? to ask general questions about people’s feelings. Would you mind…? is not used for this purpose.
Do you mind people smoking in public places? / Do you mind if people smoke in public places?
(NOT Would you mind people smoking in public places?)
We can use No or Not at all to give permission after Would you mind…? / Do you mind…?
‘Do you mind if I smoke?’ ‘No, please do.’
‘Would you mind if I looked at your paper?’ ‘Not at all. Please do.’