How do I "cap" a wire so it doesn't short or electrocute someone?
I would cut the wires to a convenient length, and then use shrink tube over the ends.  If you cut 1" of shrink tube, and cover 1/2" of the wire (leaving 1/2" of empty tube), it would offer a lot of protection.The empty section of the tube would neck down nicely...

RJR had a good idea: if you bend over the last cm of wire before heatshrinking the end, and put the heatshrink over the doubled-up wire, then the swelling at the end will keep the heatshrink tubing from slipping.

Is this PSU from a desktop computer?  If so, no need to worry about shocking somebody with the output.  All the voltages are too low.  Do be safe with the incoming AC wires, through!