Duct tape is non-conductive, so you don’t have to worry about that.
Are the inner conductors OK? If it’s just the outer sheath that’s ripped, tape will protect it for awhile (duct tape isn’t really very strong, so it won’t last a long time).
If the insulation on the conductors is nicked, I’d get the cord replaced, or spliced by someone who knows what they are doing (heat shrink tubing is the correct thing to use).