I don't think it is an accumulation of oil and dirt, but rather the steering wheel material is degrading and comes off on moist hands. I would remove it, wash it with soapy water, rinse it off, allow to dry thoroughly, prime it with epoxy primer, and paint it with two coats of paint.
Carl

1952 5-window - return to "as built" condition | 1950 3100 with a 235 and a T-5 transmission