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.

In the US university where I was a TA we were specifically instructed not to get anything worth more then $15 from a student while we are responsible for their grades. Otherwise the student or other students in the group can send a complaint to the dean and accuse you that your grading was biased.

In other countries, where customs, policies and procedures are different, the situation can be different (e.g., in Russia it would likely be fine). But if you work in the USA - I advise against accepting that offer.

The comments to my answer prompted me to add this update:

There are two issues here: an issue of personal integrity and an issue of administrative consequences. For the personal integrity, you can answer yourself: can you stay unbiased if you accept the offer? There are methods to handle it, like announcing all the policies in the start of the semester, using automatic grading with transparent standards that you cannot override, etc. You know yourself better than anyone, so you have to answer that question for yourself.

Now, the administrative aspect. Even if you know you were fair, someone can suspect and accuse you AND your student of cheating the system - and that's the issue of consequences. Consequences will depend on the local laws and policies and practices of your institution.

I personally think it could be managed, but many others would disagree, including some people with direct authority.

I think it would be wise if you did a bunch of checking before you take this on. Check with the professors in both courses as well as the department head/chair. If any of them suggest it is a problem, then it is a problem. If all say ok, then it is probably fine, as long as you have some rules for yourself about what you can and can't do.

One or more of those people may ok it but suggest some rules as well.

You don't want to do anything improper of course, but you also don't want anyone else to think that you might be doing something improper.