Would it be rude or helpful to ask a foreign coworker if they'd like for me to correct their English when there are obvious mistakes?
It is hard to say what your colleague would feel since I cannot predict how anybody would feel in any situation. Yet, if I received an offer like this, I would most certainly not get offended. I often consult my native speaker colleagues about my writing and get useful feedback. 

One thing you definitely want to do is to not do this in front of others. Doing it in an e-mail privately would certainly decrease the "embarrassment" if there is any.