Can departments sanction students who do not take sufficient precautions to protect their work?

Can a department in the US punish a student for helping other students cheat even if the student is not taking the class where the cheating is occurring?

I don't know what you can do legally, but morally I think it would be wrong to try to prevent the student from making his/her coursework from past years' courses public. Making the work public could help the student market themselves to prospective employers, and has multiple other benefits for both the student (ego boost, impressing friends, fulfilling an altruistic desire to provide a useful internet resource they worked hard to create, gaining valuable experience using github, and more) and the internet (people from all over the world could download and use the code). Who are you and your colleagues to say that your selfish needs of not having such code available for download online because it facilitates cheating by your colleague's students are more important than the needs of the student and others who can genuinely benefit from the student's work? I think your premise that the student is doing something wrong is simply incorrect.

If so, what should be done? If not, might there be anyone at the school that can do anything?

Yes, something could be done; you and your colleague could free yourselves of the harmful mindset of trying to control what your students are doing with coursework they worked hard to create after they finish taking your classes. As Massimo Ortolano suggests in the comments, stop taking the easy way out of giving the same assignments year after year and then blaming others for how easy it is for your students to cheat, and instead put in the work to give original assignments.

Actually, yes, a department can sanction a student for posting his or her work online. But a better question is should they, and under what circumstances.

For situations such as this, I think the intent of the student is essential to know. But it is difficult to learn the intent. It is possible, of course, to speak with the student about intent, but if it was improper intent you aren't likely to learn much.

As other posters have said here there are a lot of valid reasons for students to post their own work online, so, in general, it is probably a mistake to forbid it, though it is possible to make online posting a general issue in the institution's or the professor's published rules.

However, the best solution is to use assignments for which online searching don't help enough to make the effort worthwhile, as others here have also said.

One possible, even recommended, solution is to allow the student to use any sources that they can find, but to cite the sources precisely, whether it is wikipedia or anything else. This can have several beneficial outcomes, including having students learn about proper citation.