Higher is better BUT given the resistivity of your cell due to the electrolyte, you're going to get to a point where you can't go any higher because the cathode has come out of the water. There are limits. You're just going to have to play around to see what those limits are at 15ma.
The thing is though, you don't want the cathode so close to being out of the water that evaporation or a little mechanical vibration disconnects it from the cell because then you'll have absolutely no idea what you actually made as far as PPM.
We use a thin 24 AWG (american wire gage which is a standard) length of copper wire. The trick with the cathode is to minimize the surface area in the water. For a thick wire, that means just glancing the surface which isn't such a good idea. For a thin wire like this, for sure you'll still have a good half inch in the water when you're up around or over 12V. 12-15V is good. You don't really have to try to get any higher. If you're stirring though you may be able to because stirring causes the electrical path from the anode to cathode through the water to get longer (an arc is longer than a straight line, right?) but don't push it too far. If its good enough, its good enough! Words to live by. "Perfection" is not required - just good enough (wink).
As far as the power supply, does it have constant current and constant voltage settings? If so, I'd doubt you can use both at the same time. If its a current "limiter", thats different. Thats usually there for protection so if whatever you're powering shorts out, your power supply doesn't smoke or die.
What you want to do is set the voltage high (say 20+V if you can) and adjust the current limit just measuring the current across the supply with a voltmeter to get 15ma. BIG WARNING. For a power supply, I'm sure the current limit can be set to several amps at least. TURN THE CURRENT LIMIT TO ITS MINIMUM SETTING (lowest current) setting BEFORE connecting the voltmeter or you may well blow it out. Then gently adjust it to get where you want. Given your cell will be 12-15V, if the supply is outputting 20-24V, there's no way, unless the cathode comes out of the water, that the current won't remain constant at 15ma.
Let us know how you make out.
Oh, and if you manage to damage your voltmeter on a current scale, don't toss it. They ALL have an internal fuse or two you can replace and chances are good you sent it to that big fuse round-up in the sky. Some meters even have a spare fuse hidden inside but usually these are the more expensive ones. Just get a replacement fuse with the exact same rating and you're good.