Oh, Basil. What to do with you. I let it rinse under the faucet, if there's some gunk on it, like with the thick crap of Kiehl's from today's shave, I put a little soap on my finger tips and gently squeeze them off the blade's edge (the thumb and forefinger bit). Rinse. Dry on towel, stropping on towel, towel functions as cover for thumb and forefinger to ensure edge is dry.

For longer term storage (i.e. not being used the next day) or for humid conditions (such as the last two days here in Minnesota) I will simply use plain old mineral oil. That's it. Why get fancier? Cotton swab (q-tip) will handle it. In fact, I have been using the same side of a double-ended cotton swab for over two weeks, lightly swabbing along the edge of the blade. Many times I don't bother to dip it in the mineral oil container because I can get enough excess oil from another razor I had oiled earlier. All it takes is a very thin layer, no oxidation happening.