This introduces a lot of problems, all of which are difficult to solve and not directly related to the fundamental vision of Coursera. If I cheat under the full name of my sworn enemy, I can have them publicly blacklisted from any employer who uses this list. Identification seems like the logical next step, but it's no easy task especially when your students come from hundreds of different countries with varying forms of ID.
If you're serious in your suggestion, you're going to have to explain how these kinds of problems can be solved. You're then going to have the same problem Google+ had with online identities. Why should one be forced to give their real identity if they're simply trying to learn, not to be recognized for it?