Can I work as a tutor on F1 visa?

Hiroko writes, "I am a student in the United States and I have an opportunity to teach Japanese to a business executive being sent to Japan to work.  His company plans to pay me and has asked me to provide my Social Security number and complete the I-9 form.  Is it okay for me to do this?"

Actually, no.  It is illegal for an international student on F-1 visa to take employment outside the campus.  You do not have a work permit (authorization for employment) and while I am surprised that you have a SSN, but if you somehow do, it should clearly say that you are not authorized to work without proper DHS permission (it is also possible that the company may try to verify your SS# and will find out that you are not authorized to work).  Accordingly, if you want to follow the law and do the right thing, you should decline this job offer.  By the way, if you are caught, your visa maybe cancelled.