If you have a salted potato (rescued from a salad perhaps), I would not do the experiment indoors, nor would I stick my face close to it.
There are ways to kill someone with potatoes, but they don't involve electrolysis.
How, without breaking the skin?
I suppose you could burn yourself if you deliberately wrapped the wires around an appendage and immersed it in the salt water. It'd take a lot of current, and whether that would do more harm than just drinking the salt water is questionable.
The cathode reduces H+ to H2. Hydrogen gas is mostly harmless if breathed, but if you collect it, it’s explosive.
Water minus that H+ that turned into hydrogen gas leaves OH- behind. Concentrate that enough and you have a base that will dissolve your skin.
The anode can oxidize water to oxygen, which can be a fire or explosion hazard but is otherwise fine. (A stoichiometric mixture of hydrogen and oxygen is Bad News.) But this is not the major reaction because…
The anode can also oxidize Cl- (the chloride in your salt) to Cl2: chlorine gas. In the absence of considerable care (which you don’t have in this hypothetical potato salad or glass of saltwater), that chlorine gas will bubble right out of your little reaction, where it will end up in your eyes or your airways. And it will dissolve into your tears or the fluids in your airways and chlorinate you. You do not want do this. NIOSH says that chlorine gas is immediately dangerous to life and heath at 10 ppm, and you can easily achieve that in a cup with a little power supply. (Like I said, a hundred watts or so can chlorinate an entire swimming pool. Chlorinating your cornea is trivial in comparison.)
This is a major industrial process, which you should not do in your house: