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In my department, we typically adopt textbooks that are very refined, sometimes in their 7th or 8th edition. Because these books have been around for so long, all of the homework problems have been worked out in full detail, and are easily found online. So it seems pointless to assign and grade homework since student can so easily copy the answers instead of working them out themselves. But if we do that, it seems like we're then eliminating a very important learning activity.
Many of my colleagues address this by not giving very many (if any) points for homework in the final grade. They sometimes give quizzes (which takes a lot of class time) to check whether the homework was absorbed. One feature of homework is that it is done outside of class, so that the class period can be used to discuss new material. I'm concerned that taking class time to force the issue of actually "doing" the homework will detract from the other aspects of the course.
I still think that mathematics is not a spectator sport, and most of the learning occurs when the students struggle with a good set of exercises and eventually find the answers on their own. This is especially true in courses that require proofs. I've explained to my students that if they short-cut this self-discovery process, then they're limiting their learning opportunities. Still, many students copy answers.
So, in this age where all information is so easily accessible, how should we respond with how we assign homework so that students will produce their own ideas rather than copying the work of others? How do we re-enforce the idea that learning mathematics is learning how to find answers rather than just getting a copy of the answers?