Exactly how much programming will I be doing if I work as a hardware design engineer?
It's always useful for a HW engineer to be able to write code, not necessarily in the sense of architecting some grandoise software application, but it's extremely useful to be able to sling scripts around. Python and Tcl (plus bash) come to mind -- lots of tool-flows have scripting interfaces, and they often leverage these two languages. Will you be writing the best, most performant, algorithmically pure code? No. Will you sling something together that won't win a beauty contest but end up saving your team hours in the long run (assuming you don't have a tools guy)? Heck yes.

Personally I have a bit of a software background, so I'm happy to sling around C, Python and Tcl. When it comes to designing boards with SoCs and such, it is very helpful to have a blending / cross-over at low-level  where the HW guy knows enough to work with that code and modify it as necessary.

Specifics of course will vary based on your job title, but I feel 100% comfortable saying knowing how to write basic scripts in Python or bash can only help you in your career. That knowledge can be picked up in a couple of hours from some great tutorial websites, assuming you've got the basics down (don't most EE degrees now at least require CS 101?).