It depends on the question. Sometimes the question is of the form "Is this possible?" or "Is this a good idea?" Such questions are strictly answerable with "Yes" or "No" but that's not a very good answer to even a blatant true/false question. An answer in the negative is a perfectly good answer in this situation if expanded and justified.
Other questions are not asking whether the approach they are using is a good idea but asking for help on how to do what they are currently doing. This is where it gets tricky. If upon review, you feel that the approach they are currently using is seriously flawed, then a possible answer in the negative is "Your question only makes sense if you do it that way and I don't recommend you do it that way." A simple answer in the negative, even with explanatory text, doesn't really answer the question as stated.
To handle this situation gracefully, some people simply provide some direct answer to the question and then qualify it with their reservations and the answer in the negative. This gives some respect to the asker by acknowledging the validity of their literal question but allows the answerer to complement the answer with an alternative viewpoint. This is more work but more diplomatic. A answer in the negative will also likely get more credibility if it includes a direct answer to the literal question.
But there are even cases where the asker's question is based entirely on a misconception and an answer that exposes that misconception politely and goes on to try to address the real question that we think the asker is asking is a good answer, even though it is still an answer in the negative and doesn't answer the literal question.
Finally, what about other people's negative answers? How should we react to them? Well, there are good negative answers and there are bad ones. Bad ones disrespect the asker or the asker's point-of-view, don't justify their answer, or attempt to ram what is an opinion down the asker's throat.
If it's not an answer at all or abusive, then flag it. If the answer is terse and holier-than-thou, then maybe downvote it. If it's negative answer but one that might help the asker, then it's candidate for upvoting. Reinforcement from votes can help the asker come to accept the "bad news" about their current approach. In any case, as long as it is an answer, then judge the answer based on its merits like you would an answer in the positive.