not an actual answer (it would not let me post a comment to first post), but some info:
same happens on Debian, wine 1.6 from sid, official Steam instaler (.msi)
It installs fine, it opens fine, I log in, and then... it closes without any kind of message.
this is the console log: http://pastie.org/8530710
PS: almost same (crash after login) with steam 1.4 from stable. I thought updating to 1.6 would fix it, but it seems it doesn't.
PPS: a quite actual answer: updating to latest (wine) GIT release could help (I saw that on winehq regarding last Steam update, or something.... ). but I'm too lazy to compile a 32 bit package on a 64 bit machine, only to find it still not working....
EDIT: http://bugs.winehq.org/show_bug.cgi?id=35030
it IS a bug, and it has been resolved in last git release. now, if anyone wants to update...
oh, and btw, the bug is present in ALL recent (stable) versions of wine (1.4 ; 1.5 and 1.6) (wine 1.7.X is still in devel)