Both of mentioned reactions are governed by movement of protons in electrolyte. If pH of electrolyte is more than 7, then OH- will be in majority and will rule the reaction. Typical water splitting reaction in basic medium is as follows:
at cathode (reduction)
2H2O + 2e --------- 2OH- + H2 E = -0.8V vs NHE
at anode (oxidation)
4OH- ---------- 2H2O + O2 + 4e E = 0.4V vs NHE
I am wondering if we can find out which ion (proton or OH-) movement in electrolyte will govern the electrolysis at pH = 7 where both ions are in equal concentration.