It depends on which side of the Atlantic you are from. In British English, the standard response to ‘How do you do?’ is ‘How do you do?’ In India, when someone says ‘namaste’, we respond by saying ‘namaste’. The same is the case with ‘How do you do?’ This is an expression that is normally used when you are introduced to someone; once you have been introduced, you never use the expression with that individual again. The British make a distinction between ‘How do you do?’ and ‘How are you?’ The Americans, on the other hand, do not always maintain this distinction. It is common for Americans to say, ‘Fine, thank you’ in response to ‘How do you do?’ In India, you will be better off doing what the British do.
Source: “The Hindu” – Know Your English Column – Feb 19, 2008