While you need a certain voltage to overcome the electrode potentials before current will flow in the desired direction, and more voltage to get enough current flow due to solution's resistivity, *ALL* electrolysis reactions are current driven.
The simplest way of managing an electrolysis cell is with a voltage limited constant current power supply, then, for unattended operation, use a window comparator monitoring the voltage across the cell to detect significant changes of the cell voltage from its normal operating point and sound an alarm.
The poor-man's way of managing it with a constant voltage PSU is to add a ballast resistor in series with the cell to limit the current. One or more incandescent bulbs in parallel make excellent ballasts as their cold resistance is typically ten times less than their hot resistance so the voltage drop is small unless the current rises significantly.
I'd go with several 12V
universal halogen bulb sockets in parallel, then fine tune the current by plugging in different wattage bulbs. Do check that the PSU can power up and light all the bulbs with the cell shorted without tripping, which usually means you cant use all the PSU's wattage as a reserve is needed for the startup surge.