Brushed steel in the kitchen, particularly with refrigerators, is very popular, and it looks quite elegant. Unfortunately, it is also a surface that shows residue from human touch and even water very visibly. So if you use a damp cloth to clean it, the residue from the water remains. Very frustrating! There are steel polishes that work well(I recommend the product Wow!) but once you start with one it’s hard to stop because the oil or wax from the polish itself becomes very difficult to ever remove entirely.
A simpler solution is this (here’s a trade secret!): use a clean damp cloth with water and be sure to wipe it dry vigorously against the grain. That way, the residue from the water has much less contact in the brushed steel’s micro-grooves to cling to.