Since my question has been rightfully marked as duplicate, I will copy-paste elements here as an answer. Because despite being originally a question, it provides details on why do I personally think downvoting questions is harmful, including psychological aspects, and including the lack of information in the about page and FAQ. For those who have already read my previous question, you can skip everything but the last paragraph, which is new.
My personal conclusion is that downvoting questions makes the Internet a worst place, without any benefit for the SE websites, their users, or any random visitor. My rationale is the following:
Votes on questions don't affect ranking: While the votes on answers change the order in which they appear, a question always stays on top (obviously, since it is the only one). In addition, AFAIK (correct me if I'm wrong), it doesn't affect their ranking in search results, either internally in SO or externally in Google ranking.
Off-topic questions have to be closed (hum... on hold), not downvoted: There is already a process to handle questions that do not belong to SO, and this process is not downvoting, it is closing. Downvoting is only used for, according to the tooltip, questions which do not show research effort, are unclear, or not useful.
Usefulness of question is hard to evaluate: While being clear and showing research effort can be objectively evaluated (well, even being clear is pretty subjective, you can find a question unclear only because you are not familiar with the technology), being useful is seriously something you can't evaluate. How can one pretend somebody else won't have the same question? The fact that already one person had this question makes it fact likely that another person can have the same question. And even if it is not useful, how could the OP possibly know? When you don't know (and the OP doesn't know, otherwise it would not have asked the question in the first place), you have no idea if it is just a stupid silly missing semi-colon, or a more in-depth issue that may be useful to somebody else. Hence, sanctioning the OP with a downvote for "not useful" is both rude and hard to justify. Thus, so far, only downvoting for not clear and no research effort makes sense. For the following, I use "bad question" to designate a on-topic question, but unclear or not showing research effort.
Downvoting a bad question does not make it better: only a comment or edit does. Then, downvoting is not beneficial to SO in a content perspective.
Downvoting a bad question increases its chances to be incorrectly closed: often, I see bad (mostly unclear) questions being closed solely because the negative counts attracts compulsive closers, while the question was not unclear to me, and I would have otherwise proposed an answer. (In this cases, I shortly answer as a comment...)
Good questions are already encouraged: A good question is both "rep-encouraged" (a good question gets upvotes and increase rep), and "naturally encouraged" (even without rep, a good question is more likely to get a good answer).
Bad questions don't have to be "rep-discouraged": One may argue that downvoting makes one lose rep, and hence allowing to downvote questions discourages asking bad questions. This is redundant, since anyway asking bad questions is already naturally discouraged by the fact the OP won't get good answers to bad questions. Note: the difference between encouraging a behavior and discouraging the opposite behavior is subtle but exists. In the case of answers, we want both to encourage good answers (of course), but also to discourage bad answers, since they are harmful for the OP, but would be encouraged otherwise since they would potentially accumulate rep, and are not "naturally discouraged" (Hence, there would be nothing to lose to post a bad answer if the downvote system didn't exist).
But despite this apparent uselessness, I do see a very harmful downside: potentially hurting psychologically the OP, and eventually turning them away from SO. Not only this is not beneficial, but is harmful both for the OP, in a human point of view, and for SE, in a business point of view. Why is this? Even though, ideally, a downvote would tell the OP: "a member feels that your question is unclear or does not show research effort, and suggests you improve it", the OP is not aware that this is the intended message. Only experienced users know, but anyway experienced users do not post bad questions (well, I hope so ;-)). Proof: this intended message is not written anywhere in the about page or even the FAQ.:
The about page does not mention voting up or down questions.
The FAQ extensively discusses what are off-topic questions, but does not describe what is a bad question (on-topic). It merely gives advice about how to ask a good question, but never specifies you will be downvoted if you do not follow the advice. Specifically, I haven't found (maybe I am blind) a section explaining In which case should I downvote a question? (for voters), and What does it mean when someone downvotes a question? What should I do about it? (for OP).
Then, the only information about this is on the tooltip that appears when you let your mouse hover on the down arrow, which I don't expect a new user to do (seriously, do you expect new users to hover and wait 5sec on any interface element before asking his first question?), and even if he did, it still isn't clear about the intended message. Then, when receiving a downvote, the only solution for the OP is to guess what does it mean. Personally, I would guess something like: "Your question s#$!#, you n00bs, stop programming now, you don't deserve our help". And none of us can be aware of who is the OP, what is his life, is he fluent in English, did he actually try hard to start programming by his own, to ask the question, how important is it for him, and is he psychologically fragile or not?
The accepted answer here states that the benefit is that it provides a visible hint. Actually, it only provides a better visible hint, since a hint is already given by good questions having a "more positive" score. So SO allowing downvotes implies this increased visible hint is worth the psychological harm caused to users. I personally think it is not worth it, but that is up to the developers, and the SO community, to decide.
