Following the Steam support page do:
1. Install mesa-utils for i386:
sudo apt-get updateAfter that you run a diagnostic test:
sudo apt-get install mesa-utils:i386
glxinfoCheck on the top of the test results-> direct rendering: Yes (should be)
Check are you a part of 'video' group:
Reinstall all video drivers - open Synaptic, refresh/reload it and find all packages starting by 'xserver-xorg-video-*'
Mark all the video installed packages and reinstall theme, then reboot your machine and check Steam again.
It what I can do.
If you still have a problem, please contact the Steam support.