Ugh yes, I can relate to having these types of members on a forum. And yes, it does make other members feel that they want to leave to not be around them, but I can't just ban someone because others don't like them. On Christianity Haven, I'm very lenient with our rules. I want members to actually learn why what they're saying is a rule violation and then to learn from it and change. It can take years for that to happen! But I have seen some members improve (as long as they stop drinking).
I've had my fair share of them. I usually PM them to drop the negative vibes or ask them to leave if they can't change their mindset. It's important to deal with this quickly before any other members get offended and leave your community.
I never had to deal with this on any of my communities thankfully, but I would send the member a private message asking them directly how I could help make their experience on the forums better. Then if they continue to complain, I would kindly ask them to leave if they don't like the community, before they start drama or breaking rules.
Unless they are harassing users are targeting particular user, or breaking any of the forum rules, I do not mind too much negative member on my site. In fact, having user who criticizes almost everything will also bring more activities in the forum.