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.