I live in the tropics in Thailand so i have to cope with high humidity. After I use my razor I wrinse under a cold tap (don't have hot water) then strop a few times on a towl to dry. Then I do a few laps on the strop to make sure the edge is dry and clean. Then the razor goes into my basket holding all my shaving gear and strop which gets put into my room away from little fingers. I never bother with oiling my razors unless I am going to be going away for a week or two and won't be using my razors.
But where ever you decide to store your razors just make sure they are in a dry airy place.