How can I save my DirectX choice in Saints Row: the Third so it stops asking me about it?

Replace the launcher with your chosen game executable. A summary of the steps here:

1. Copy the Saints Row .exe files in your Saints Row the Third install directory to a different folder (that should be game_launcher.exe, saintsrowthethird.exe, and saintsrowthethird_dx11.exe)
2. Copy back the one you want to always use - saintsrowthethird_dx11.exe to use the DirectX 11 version, or saintsrowthethird.exe to use the non-DirectX 11 version.
3. Rename the file you copied to game_launcher.exe.

This tricks Steam into directly running the version of the game you want, instead of putting you through the launcher.

If you go to the folder in which SR3 is installed their is a exe for each DirectX version if you run that exe it will load the game. Just simply set up a shortcut for that exe.