While I understand that in the post Wikipedia era it's generally more difficult for professors to come up with homework questions whose answers don't exist on the web, that's a problem for the professors, not us. Adapt or die applies.
My teaching experience is extremely limited, and at a level a tad lower than University, however from that limited experience my conclusions are that cheaters will cheat, and there isn't much you can do about it. I was perhaps extremely lucky in that the majority of my students were interested enough in the subject matter to not cheat, even though I actively encouraged them to use Wikipedia and Stack Overflow (Programmers didn't exist at the time) and I'm very proud that some of them are today colleagues. A couple of the cheaters also made it, but most of them are flipping burgers somewhere. Oh, well, c'est la vie.
But, I only taught for three years, let's see what someone a tiny bit more experienced has to say:
My opinion is that there is nothing wrong at all with posting homework questions here, particularly interesting ones, and I find much of the negative reaction to homework-question posters to be somewhat strange, alien to my way of learning mathematics in a give-and-take exchange of mathematical ideas. Surely posting questions here and studying the answers is not much different than studying hard in the library, talking mathematics with one's colleagues at math tea or talking to one's professor, which are all excellent ways to learn mathematics. In particular, I expect that students who post questions here might learn just as much if not more from the resulting answers as from their professors---we have a number of talented mathematicians, who are very good at explaining things---and that math.SE provides a valuable service to students having unapproachable professors, having professors who do not explain well, or who have few colleagues able to help them. Furthermore, the math.SE community strongly benefits from the questions and the insightful answers that might be posted.
So my opinion is that there is no homework issue to speak of.
In particular, I hereby give all of my own students complete permission to post any and all their homework problems here, and indeed I encourage them to post their questions here and to study the answers well and thereby to learn some mathematics. I will be testing them on their understanding at the exam.
I would also encourage all mathematics professors to adopt a policy of encouraging collaboration on homework among their students, as talking about mathematics with one's colleagues is assuredly one of the best ways to learn mathematics. Indeed, I recommend that all professors should actively encourage their students to form study groups in order to work on their homework problems together. Learning as a group, they will go very far.
Programmers already has very strict policies regarding the quality of questions, perhaps stricter than most other Stack Exchange sites, and there's absolutely no point at all in coming up with a specific policy regarding homework questions or what appears as homework questions. Any question that's within the scope of the site, as defined in our FAQ, and shows sufficient prior research is welcome to the site.
Lastly, and only for completeness sake, while "professionals" is in the description of the site, it does not disqualify amateurs, enthusiasts and students from using the site, provided they conduct themselves in a professional manner. Our goal is to build a high quality canonical resource that will hopefully help current and future professionals, we don't check credentials at the door, and we never will.
On Meta Programmers:
On Meta Stack Overflow: