I am the manager of a small team of software engineers. One member of this team write very insecure code. I have tried showing him documentation like the OWASP Top 10, but he just doesn't get it. I can't knowingly put vulnerable code into production. So what do I do with this programmer?
You could just open a new bug report each time you see a security problem in his code. Eventually he will get the idea, or at least learn a thing or two...
Communicate to him. In writing
Establish written coding standards. Including security.
Include fixing the problem in his performance reviews.
Explain to management the risks/costs associate with violating.
If he violates, refer him to management/HR
Or, if you don't wish to get rid of him for some reason, stick him on tasks that are not security sensitive - test writing, utilities, code prettyfying, styling, etc...
Also, please don't literally follow your own answer ("just fire him"). Without following points #1-5 you'd be opening yourself to lawsuit risk etc... - even though that (firing) might be the desired outcome.
Train them. As you have a leadership role in your development team it is your job to assist in bringing developers up to the standards that are expected of your team. If you've already invested a large amount of effort into this person and they have achieved no success and aren't clearly focused on attempting to do better. At that point I think it is time to consider letting them go or moving them to a more junior level project that is fitting of their ability.
Have you considered doing code reviews with your team?
Don't single out this particular programmer, rather review everyone's code. That way he will learn from his peers how they code for security, see examples of how they do it, and learn why they consider it important.
If after doing this for a while he does not come up to a level that the other team members can respect, it will become obvious to all and you will not be viewed as heavy handed or the "bad guy".
A side benefit of code reviews will be higher quality, more secure code from all members of your team.