I realise this has been a long standing problem, but it still occurs
in the most recent version of Crossover (13.1) on Mac OSX.
Steam crashes after it finishes "checking for updates" , I have
tried removing Steam and reinstalling it into the same bottle, but
the problem still persists.
I am running Mac OSX Mavericks 10.9.1.
Actually, it's not really a long standing issue with CrossOver. What you describe was breakage that occurred when Steam rolled out an update that broke Wine variants across the board. It was fixed for CrossOver with version 13.0.1 (less than twenty four hours after the breakage). Maybe you could tell us a little more about your system?
Your account doesn't show an entitlement so I have no way to place what version of CrossOver you might be running and if perhaps you upgraded from an older version (which rarely causes problems but I would like to rule it out).
There are a few things that could be going on with your system.
Do you have a third party input manager on your Mac? (like Razor Synapse or 1Password)
Have you tried installing Steam into a fresh bottle?
Have you tried removing the Installers folder so that CrossOver would be forced to download the lastest installer of Steam?
Can you give us more details about your system? When did you move to Mavericks? Did you use a different version of CrossOver before?