The most recent Cataclysm Beta build has added more details about how the Upscale Resolution Texture system will likely work for Cataclysm Classic.
As you log on to Beta, there's a new button on top of the Addons one in the character selection screen. This button guides you through enabling HD Textures for Cataclysm Classic.
Clicking on it will close the game and lead you to the installer options for Cataclysm.
There, you will be able to (in the future) download the assets required for HD textures.
Currently, these settings are not present in the Battle.net client, but they will likely look similar to the same settings in Diablo IV:
Once the textures are downloaded, which is around 30GB right now, you will be able to enable/disable them from the in-game client. You aren't required to always use them after downloading.