Do you have to use your full name to enroll, in the field asking for it? I've probably become excessively paranoid about giving personal info on the web - Stanford is probably OK...
Judging from my account, no you don't. Maybe if you are planning on being certified but just to take a course, no.
Coursera is trying to branch out into academic partnerships and professional credentialing, which will probably want your legal name if they become relevant to your use of Coursera. I don't think they have a way to confirm your name for a simple course enrollment.
Only really matters for certificate.
The signup form has a field named Full Name, but what's stopping you from using a pseudonym?