Is it safe to touch a running 1 hp electric motor?
What do you mean by "touch the stator"? The Stator is a set of coiled wires inside of the frame of the motor. You should not be able to 'touch" anything electrically live without disassembling the motor frame, in which case it cannot be "running". Your question either makes no sense or you don't understand what you are looking at.

If you can see the coils and possibly get a finger through an air flow opening, besides it being exceptionally stupid to stick your finger into a piece of rotating machinery, the coils themselves are insulated, even though they may APPEAR to be bare copper. That however does not mean it is electrically safe. There could be leakage due to age, damage, defect, etc. You don't want to test it by stopping your heart...