There are two types of control of what reaction may occur: thermodynamic control and kinetic control.
Thermodynamic control dominates in near-equilibrium conditions, which in case of electroplating means small voltages and relatively high concentrations.
When applying high voltage, metal ions in near-electrode space are quickly exhausted and we move from equilibrium conditions to kinetic conditions when the first reaction that may happen occurs. In water this is oxidation/reduction of water.
The main criteria for choice of the voltage in electroplating is so that diffusion of metal ions from solution happened fast enough, but electrolysis of water and/or oxidation of anions didn't happen. This means low voltages, preferably below 1 V when possible. Electrolysis of water may happen if voltage applied is above 1.23 eV, so exceeding this value is allowed only if the procedure and experimental setup is proven to work.
Why electrolysis of water in this setup bad? Leaving aside waste of energy, it creates bubbles of gas stick to the electrodes, resulting (at the very least) in inhomogeneous plating. Also, mind that the transport of ions in water is a diffusion process and electrode potential does not have that much effect on it.