Why are electrolysis cell plates turning a brown color?
Since the carbonate radical is covalently bonded, it probably is not being decomposed electrolytically. Much more likely is that iron in the stainless steel is being removed and oxidized, leaving a thin film of "rust" on the plates. 

A few ways you can prove this:


Immerse the browned plates in oxalic acid, which will slowly dissolve iron oxides, but leave carbon untouched.
Substitute platinum electrodes -- admittedly expensive, but Pt is the "gold" standard for electrodes. They should not discolor.
Spray the browned plates with a solution of potassium ferricyanide. If free iron is present, a blue color ("Prussian blue") appear.