Is it a safe voltage to use?

12 volts won't kill you, but you will feel a slight shock from 12 volts. The input of 120 volts is standard output of a common household outlet so you will feel that a lot more, but again 120 volts isn't going to kill you. I would simply cut off the power supply while handling it if I were you.

This would be completely safe. However, when dealing with any electrical source, be careful and use safest methods, if nothing else, to get used to the practice. Typical skin resistance is 100,000 ohms so current through the body would be limited to 0.01 to 0.02 mA. 8 mA is considered a maximum safe limit of electrical current passing through the body. I won't recommend it, but I have put a 12 volt battery across my tongue and I could definitely feel a tingle becuase the resistance of your tongue is far less than typical skin resistance (and here I am preaching about electrical safety!).

12v is low enough that contact with dry skin wont give you a shock. The lowest reported voltage which has caused electrocution was 32V (DC) electrocution when wet with water. Generally you wont feel a shock with dry skin below about 50v.

Alkaline water is conductive. As your both hands were wet with this conductive liquid, your skin resistance falls down as low as a few hundred ohm. Therefore,even with such low 12V power source,you might get electric shock from it or even quite harmful.

Current passing your body could be .....

current = 12V / 500 ohm = 24mA

Good enough to kill someone.

WEAR A PAIR OF PLASTIC GLOVES MIGHT SAVE YOUR LIFE.

Alkaline water filters are pseudo science. Don't waste your time.