If the bowl is wooden, covering it wet may lead to the lid warping. I personally lid most of my soaps (since they're usually not really wet after lathering, the brush soaks up most of the free moisture with the soap). This is more to preserve scent than any other reason though.
I go to work at an ungodly hour every morening, so I shave the night before. I usually leave the lid off overnight, then put it away in the morning.
Either method works just fine though. Putting the lid on will probably increase the longevity of the scent, although I have no empirical evidence to support that claim. Leaving it open will help dry the soap out, making it last longer. You shouldn't have to worry about the "soupy mess" unless you leave a puddle behind, though. The soap will absorb the remaining moisture, which will soften it up some (which is why it won't last as long).