Another way to fix this problem, suggested to me by a colleague, would be to have rebuild simply clear all password fields, requiring them to be reentered on each rebuild. This might be easier to implement.
This solution is even more secure that the originally proposed solution. Say that a user runs a build using his or her password. Later the user's password is revoked. The originally proposed solution of checking the user would allow the user to rebuild even though his or her password has been revoked. Forgetting passwords on rebuild and requiring them to be reentered on each rebuild addresses this scenario.
And I do consider this problem to be a bug rather a feature request. I would say the priority should be higher than Minor, as this problem exposes a significant security vulnerability. I've had to disable the Rebuild plugin on my Jenkins server as I have a pipeline that uses a password parameter. This is a significant impact to my Jenkins users.