Okay, assuming it's a multimix 8 USB...
Follow the quick start guide to install the Alesis software and drivers on your computer. Go to the Alesis site and make sure there aren't any updates--install them if their are.
Making sure the USB cable is connected to your computer, open Audition. Go to the "Edit" Menu. Right at the bottom select "Preferences" then "Audio Hardware". Under device class, select ASIO (I'm assuming the Alesis comes with ASIO drivers--if not, leave it at MME.
On the same menu, select your Alesis under the Default Input and Default Output menus.
Select the Audio Channel Mapping menu (still part of preferences) and check that the appropriate Multimix outputs are selected to the L and R inputs to Audition.
At this point, you're ready to record. You can work in either the Edit (Waveform) view or Multitrack depending on which suits you. The simplest is probably the Waveform view unless you're going to want to do punch ins and so on as you go. Just hit the red record button and set the file name, bit rate, sample rate, etc. in the pop up then do a test recording for levels. Once you have these right, go for it.
