There are (at least) two badges, which are related to a negative score.
Reversal: Provided answer of +20 score to a question of -5 score
Peer Pressure: Deleted own post with score of -3 or lower
I've read comments like this several times:
Please don't downvote below a score of -1, even if the question in it's current form needs some improvement. A score of -1 is enough to show that the question needs work, anything below that is of no use. Also, if you downvote or vote to close, please leave a comment explaining why you did so, but wait at least 24 hours after asking the OP for improvements to the question before voting to close. (by Henri Menke on https://tex.stackexchange.com/questions/150006/changing-to-a-vertical-table-in-latex#comment341490_150006)
I've even read a comment like this on a question which clearly stated that: "How to do that? I'm lazy and don't want to look for a solution." - The comment got upvoted and downvoting stopped at -2 (-3 actually, but someone gave an upvote (I don't know why)). The question was finally closed as a duplicate (as far as I can remember).
These comments usually get upvoted quickly, so I think that these two badges are outdated here. I suppose we don't want anyone to be motivated to downvote anything to make a Reversal badge possible.
Are they outdated here Are the badges the same on all SE sites? Shouldn't they be switched off (moved to inactive state)? Or: is the idea of not voting below -1 contradicting with original SE intentions?
(Should this question be tagged as [support] because "I need help with the use of one of the site's features"?)
About answering downvoted questions:
Should we answer questions which are downvoted / low quality?