According to: https://hudson.dev.java.net/changelog.html#v1.180
Hudson displays the password textfield for a user when the "Hudson's own user
database" is enabled. When I upgraded to 1.181 this was not the case.

My Hudson System Configuration:
Enable Security = On
Hudson's own user database = On
Logged-in users can do anything = On

Using the above configuration, the password textfield was visible in version
1.177. I have not tested this issue on v1.183.