Probably anybody visiting this site regularly noticed that quite often there are posts which are not at MO-level and quite frequently also of very low quality. (Many of them are simply copy-paste of an exercise or an assignment. Users with sufficient reputation can find some such posts in moderation tools or in close vote review history.) And there have been some complaints about these posts here on meta, for example: Proposal to more aggressively delete off-topic questions or Can the clutter of obviously off-topic questions be removed faster?
Some time ago, Mathematics StackExchange started to require registration before asking a question: Let's require registration to ask a question. (Unregistered users still can answers questions and make edit suggestions.)
I do not have any data which could quantify to which extent closed/deleted questions on MathOverflow are posted more often by unregistered users. (And I do not know whether and how such data can be obtained.1,2,3,4) However, judging by the off-topic questions I have seen, this is quite often the case.
I suppose that users who did not bother to read in the help center what is and isn't on-topic on this site, are more likely to choose a site which does not require registration. (If they know about the existence of two sites at all.) So my guess is that requiring registration might help reduce this type of questions. (At least the ones coming from unregistered users.)
Of course, there might be also negative consequences to this change. Maybe som unregistered users will simply post their questions as answers. And probably some good questions have been posted by unregistered users - this would be no longer possible.
TL;DR: Should MO require registration before posting a question? Would this help to decrease frequency of off-topic questions? What are possible negative consequences?
1There is currently one user which is a moderator on both sites. The mods have access to some data and tools which regular users do not have - perhaps moderators from Math.SE are able to assess to which extent this change helped. And moderators from MO might be able to say to which extent this really is a problem.
2Some analysis concerning number of closed questions after joining the Stack Exchange network was done here: Has the switch to SE 2.0 increased the rate of low-level / inappropriate questions? However, in this case it is more interesting to know to which extent unregistered users contribute to the closed questions.
3 Based on the comments here it seems that such data cannot be obtained from SEDE. Some statistics on this were made for StackOverflow in 2011, about the time when SO started to require registration for asking questions: Stack Exchange Community Statistics results: In which ways do unregistered users participate? Looking at the results there, unregistered users were responsible for more than 10% of questions (before adding requirement to register) and about 3-4% of answers. However, it's hard to say to which extent behavior of users on MO and SO might be similar.
4In the absence of better data, we can simply look at the questions posted yesterday. This is a very small data sample - on the other hand, the period is short enough so that this will probably not be influenced too much by the questions which have been deleted in the meantime. (Users with access to moderation tools can check whether there are many recent deletions. However, there are some posts which were closed and deleted in that period - I was able to find one deleted post in my flagging history. But this was a post deleted by OP - so it is not shown in the moderation tools. Only registered users can self-delete posts.) And one-day period is also short enough to go through manually.
Of the 6 questions posted by unregistered users, 3 were closed as off-topic, 3 remained open. Among the questions posted yesterday, 5 were closed; 3 of them posted by unregistered users, 2 from registered accounts. See here for the complete list of the questions. (Of course, this is based only on the question which were are still undeleted.) All 21 answers posted on the same day were posted from registered accounts. (Again with the same caveat - possibly some answers might have been deleted by the community since then.)