To access the Public Test Server:

Log in to the Star Wars: The Old Republic launcher using your username and password.

Click the Settings button (gear) in the bottom left corner of the launcher

Under “Enable Public Test Server Access”, select “Yes”

Click “Save”

Now the bottom left corner will have two buttons: select the new button next to settings

Under "Pick an Environment" select "Public Test"

Click "Next" and you will begin patching on the Public Test Server*

Click "Play"

I'm not sure why, but depending upon what "loader" you use, you may not have a "settings" button at the lower left of the Launcher, after you log in. In that case, click on your account name at the upper right of the Launcher, and select "settings" from the drop down.

If you don't have the settings button, you will only have the "new" button on the bottom left.

I know, details, details. But I had time to wait for the PTS to download.

(about 2 hours, as it turns out.)