mIRC should only ask for the password when you attempt to open/maximize it. It shouldn't be asking when you change desktops. Unless perhaps if you have mIRC running on those other desktops and it isn't minimized.
Does this happen only if mIRC is minimized on desktop 1 or only if it's not minimized or all of the time?
I've used similar multiple desktop setups before, but never tried them with mIRC and I don't use passwords on mIRC, so I haven't run into this before. Although Windows doesn't have a specific multiple desktop setup, I am pretty sure I've seen utilities that provide that same setup in Windows. I wonder if the same thing will happen with that setup? I am thinking it probably will, and if so, that is definitely something that should be fixed. If it's only in WINE/*nix, then it would still be good to fix, but Khaled doesn't provide much support for anything besides Windows unless it's a major issue that affects a lot of people.
Also, the fact that you can't use the password dialog is probably going to be related to an issue/bug in either WINE or evilwm.