I disagree with this idea.
Someone asked a question, and someone answered this with a worse solution or not an solution at all.
If someone asks a very bad question, close it with the appropriate reason and (optionally) downvote it, depending on how bad it is. If someone posts a very bad answer, downvote it and explain the reason in the comments section and (optionally) flag it for the appropriate reason and (optionally).
You're the first to downvote and the answer/question reaches a score of -10 after that for at least X (25 or maybe 50?) times you'll be rewarded with the "Helpful downvote" badge.
I don't think we need to encourage downvotes specifically. People already downvote bad questions and answers. Giving away badges for downvoting would encourage people to create fake accounts just for the purposes of downvoting and this could be a big issue later on. Also, I personally think there's no need to downvote an answer to -10 or above. If the post is so bad, it shouldn't exist on the site -- it should be deleted / flagged for moderator attention instead.