I have a bunch of loosely related reactions that I think add up to "no, this is a bad idea".
First, the idea seems to assume that everyone who asks a question on MSE is asking it in order to get help with a homework problem; otherwise the proposed new box doesn't make sense. And while homework questions may be in numerical majority on the site, they are rarely the best and most interesting questions we have.
At one end of a rough spectrum, there are non-mathematicians asking something about mathematics they have always wondered about. Such people should not be expected to be able to "do some work" to show on their own before asking, and it would be counterproductive to ask them to. When asked well, one of these "naive" questions can generate more interesting answers than ten or twenty do-my-homework questions. If we presented new users with a question form that suggests that they're not allowed to ask questions they do not have a partial answer to already, we'd run a real risk of missing out on good and interesting questions.
At the other end, there are mathematically sophisticated users who know enough to ask about exactly what they don't know about their problem. These people shouldn't be required to "show work", either, because if they had any work to show, they'd have asked a different question. They can and should be expected to provide some context and motivation for their question, of course, but that's different from "you must show some work on the question you actually ask". One of these questions is also worth twenty do-my-homework questions.
Then, there are some actual students who're smart enough to ask about their understanding of what they're studying, rather than about their homework. They should explain their confusion as well as they can, but again this fits badly with a division into "the problem" and "how far I've come with it".
Or there are non-mathematicians who simply come across a real-world mathematical problem they need a solution for in their (otherwise not math-intensive) job or hobby. There are not many of them who find MSE, but those who do ought to be the prototypical example of an ideal question asker on a SE site.
The idea of tailoring the question form to the (on average) least interesting kind of question for the site would on one hand make people less likely to ask the more interesting, and on the other hand reinforce the (already strong) perception that homework help is what MSE is about. That's not a desirable outcome, I think.
Second, I think homework askers who "need" this the most (that is, those who don't spontaneously tell something about what they've done with the problem) would be overwhelmingly likely to write something like "I have no ideas, I don't even know where to start" in a box asking them for their own thoughts.
After all, it's a nigh certainty that this strategy would work for them -- for every homework problem submitted without comment, there are always answerers who rush to provide a worked solution. Technically forcing the question to say explicitly what is already painfully apparent without the new box wouldn't change that.
Hmm, this needs to be emphasized with a bold ordinal of its own: Third, the real problem is not that the question asking process allows bad questions to be asked. The real problem is that bad questions receive answers that satisfy the asker's needs. As long as that is the case, adding more hoops to jump through for the askers will not solve anything fundamental.
Fourth, even for homework questions, I think that demanding, specifically, work on the problem of the asker is missing the point a bit.
Hopefully we agree it shouldn't be the mission of MSE to provide a worked solution that the asker can copy into his hand-in, whether or hot the asker showed some work or not. Sure enough it feels extra annoying when this happens and the asker didn't show any effort at all -- but that doesn't mean that partial effort shown by the asker should excuse our expanding it into a full solution for him.
What we actually need is not work shown per se, but rather sufficient information that we can figure out which piece of knowledge this asker lacks before he can solve the problem himself, and then provide that to him. A partial solution is one way to provide that information, but it's not the only one. In fact, as Kasper also notices, it's a not even a good way to provide it in many cases -- deciphering the partial solution is tedious work compared to if the asker had simply written some prose about his angle of attack, from which I can see what his problem is.
Sometimes a prose description of the asker's thoughts will be so confused that I'll need to ask him to provide some worked details of what he means -- but that is okay and what comments is for; I don't think it means he ought to have provided the details automatically. All in all, I'd much rather have all homework questions contain a mandatory prose paragraph about what the asker finds difficult but not even a start of a written-down solution, than I'd have the opposite.
Sometimes, it may even turn out, after understanding what the asker's trouble is, that the most convenient way to show him what he was missing is to show a full solution. That should probably be done cautiously, though, because it short-circuits his teacher's ability to adapt his style to the student and so may do more harm than good in the long run.