This badge would be for people who have left comments in roughly 20 questions or answers they've downvoted.
It'll encourage people to explain why they're downvoting without forcing them to do so.
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My issues with this are:
It does not encourage positive behavior, you have to be a "downvoter" to even be considered.
It would be really easy to get so it would have to be bronze, but it seems like a lot of effort for a bronze badge.
It could encourage negative behavior eg "-1 I do not like your gravatar" and so forth
What happens when you explain your down vote, the user incorporates the comment, and then you give the modified post an up vote?
This would encourage non-anon voting, which is risky. If I see a negative comment on my post and the person has "Stalter and Waldorf" I can safely assume the person leaving the negative comment is the person who downvoted me.
While this is a great idea, it falls in the same way that all the "force a comment on downvotes" fall.
My recommendation would be to abolish downvoting of questions and only allow upvoting of questions: let a deafening silence (no answers, no upvotes) be the indication of a question to be ignored.
A question with no answers and no upvotes has but a marginal existence. It doesn't really worsen the signal-to-noise ratio, especially since S.O. search results will show whether a question has any answers or not. You can include a (user-toggleable) feature whereby any question without an answer would be excluded from the search results. Now that would improve the signal-to-noise ratio!
The focus should be on bubbling good answers to the top. There are no stupid questions, only stupid answers.