Wanted to add a thanks for the final, highly useful tip in this thread.
I had the same thing happen, in my case because I had some .command batch files that included sudo, and I'd accidentally closed one of the Terminal windows while this was still asking for a prompt. I'd expected it to kill the process, but instead the sudo was sitting there waiting, and locking up new Terminal windows.
I could see all the login processes sitting there in Activity Monitor--one for each window I tried to open, all of which refused to Force Quit--but didn't realize there was also a sudo sitting somewhere that was the actual problem.
Surprising that, of all the things that came up in a Google search for blank terminal windows or stuck login processes, none suggested this.