I think that most of us dry them with tissue, remembering to wipe between the scales. Avoid if you can, letting water get into the pivot, a couple of oil drops there helps. Leaving them to air dry only may allow rust spots to form.
There are razor stands available to purchase, or you could make one. Otherwise, to stand one up with the blade open, use your imagination, just avoid the pivot being the lowest point.
Obviously, if you share a bathroom, be mindful of leaving an open razor.