You have two options for running Steam on Linux : 1. using the Steam client or 2. using Wine Steam.
The first option only runs games inside the Wine bottles. The client will have the same rights as any other Linux programs.
The second option runs Steam Windows client inside a Wine bottle. Wine does not do any sandboxing so it does not add a security layer.
From Wine FAQ
7.5 How good is Wine at sandboxing Windows apps?
Wine does not sandbox in any way at all. When run under Wine, a Windows app can do anything your user can. Wine does not (and cannot) stop a Windows app directly making native syscalls, messing with your files, altering your startup scripts, or doing other nasty things.
You need to use AppArmor, SELinux or some type of virtual machine if you want to properly sandbox Windows apps.
Note that the winetricks sandbox verb merely removes the desktop integration and Z: drive symlinks and is not a true sandbox. It protects against errors rather than malice. It’s useful for, e.g., keeping games from saving their settings in random subdirectories of your home directory.
If you really do not trust Steam client, you could run it in a Windows virtual machine.