+1 don't do anything to it other than a superficial cleaning and oiling. I wouldn't shoot it, including period-appropriate/safe blackpowder, but OTOH couldn't blame you for curiosity to check it out--after a pro smith checked it out first. The .38 is .38 WCF (Winchester Centerfire) aka .38-40. The "38" part is a misnomer not uncommon in the industry As suggested, as it is closer to a .40 cal (some modern revolvers so chambered can also shoot .40 S&W and 10mm with different cylinders) . So it really is more a .40-40, as in .40 cal bullet over 40 grains of powder. Maybe Winchester felt that tripping that off the tongue sounded too close to .44-40! In any event "manufacturer's license" to call their cartridges whatever they wanted. (a .44 Mag or Special is really a .429 or if you will, .43. .44 sounds better!).
Great old west ground with decent "inherent" accuracy and pop in modern loadings/chamberings, that seems to be getting some renewed, if still niche, interest.