What is made at the copper cathode when electrolyzing water?
Look at your redox potentials for the half reactions.  12V is tremendous overkill.  At the cathode (reducing) you will make hydrogen to which copper is inert.  Could you reduce sulfate to sulfide or sulfur and have that react with copper?  CuS is black.

At the anode (oxidizing) you will oxidize water to oxygen - but one atom at a time.  It will be very reactive to the copper surface.  CuO is black.  You will also oxidize copper to Cu(II).  It coordinates four waters square planar.  The apical waters are distant and weakly bound given the Jahn–Teller effect,

http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Coordination_Chemistry/Coordination_Numbers/Jahn-Teller_Distortions 
http://www.adichemistry.com/inorganic/cochem/jahnteller/jahn-teller-distortion.html 

One possibility is to add a series resistor to lower the working voltage.