I usually use a plastic brush and a little soap and hot water. I hold the handle and run the brush down the length of the blade, brushing away from the cutting edge, never towards it, to avoid dulling the knife. If there's something really sticky on it, I may use a plastic scrubber, but I seldom need that.
I rarely dry the knife -- stainless doesn't need careful drying. Instead I just bounce the end on the cutting board to knock off most of the water, then put it back in the knife block with the sharp edge away from the wood.
I've had the same set of stainless Wusthof Trident knives since 1968 or so, and they've almost like new, and they need sharpening only once every couple of months. I do the same with my newer Hattori Gyuto knife, which is also stainless (albeit 60-layer Damascus steel). Drying these razor-sharp blades by hand is just another invitation to a severe cut, IMHO.