Later in Feb 2018 the old launcher will be retired. The new MWO Portal hasn't worked for me in CrossOver. Instead I recommend you install Steam into a Windows 7 bottle (it's a supported config). From there login to Steam and you can install Mechwarrior Online.
But when you hit play it doesn't work as it runs the 64bit version. Instead search for the 32bit version and run that.
You usually find it in
~//Library/Application Support/CrossOver/Bottles/<bottle name>/drive_c/Program Files (x86)/Steam/steamapps/common/MechWarrior Online/Bin32/MWOClient.exe
You can just double click that and CrossOver will start MWO.
In the game I've found you have to keep the video settings at the lowest for it to run with enough FPS and no crashes.
When a patch comes out you should be able to update it through Steam.