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? I have a colleague who is very new to this country (USA). His English leaves a lot to be desired, to the point that it often actually hinders communication. I think he's trying to learn, but it's obviously hard.

Sometimes in an email, there will be a few mistakes that are fairly glaring, even if I get what they're trying to say. I was considering asking him (so as to not just give unsolicited advice), "would you like me to correct any errors in your writing I see when you email me? It could help you improve more quickly." 

It could definitely help him learn, but I'm also aware that it could be embarrassing for him, as well as presumptuous on my part. Personally, I (like to) think I'd really like it if someone made me that offer if I were in that position, but of course I can't know for certain.

Is that a good idea, or is it better to just mind my own business?
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.