The response to "How do you do?" is "How do you do?"
Except in its formality, it follows the convention of "Hello" and "Hello." It is not "How are you?" in which case you can offer a "Fine, thanks."
When you offer anything other than "How do you do?" you're showing your ignorance (note: not stupidity, just lack of knowledge) of the convention.
Saying "Fine, thanks" makes it a one-way conversation, all to your benefit, and you do so at the risk of the other person's taking at least slight offense.