The 'would you' form is more polite. It's commonly used.
'Do you mind' sounds more like a command, whereas 'Would you mind' sounds like a request.
'Do you mind' can also be used as a criticism of someone's behaviour - 'do you mind not smoking while I am eating' would normally be said by someone who is quite cross about the smoking. It can be almost sarcastic in use.
I wouldn't recommend using that sentence ("Do you mind opening the window for me?") in order to make a request. It doesn't really sound like a request and could well come across as rude.