My student in one course asks for paid tutoring in another course. Appropriate?
You're correct in feeling uneasy.

Even if you are not yourself improperly influenced, there would definitely be the appearance of possibility of improper influence, which is as important, in terms of the impact on other peoples' actions, as actual impropriety.

Even tutoring _for_free_, since you are in the same dept as the people giving the other course, could be perceived as improper, for various reasons.

Best to keep "squeaky clean", I think.