This issue comes from the food service industry. It depends on the type of cup/glass. Plastic glasses used in like restaurants have the little plastic strips that prevent the glasses from sealing against each other, allowing water to drain and the glassess to dry while stacked in the cup down position. Coffee cups are normally stored in flats, cup down, and not stacked. Food services are required to store glasses and cups cup down to keep them clean and dry.
What you do at home is up to you. I hate a musty/moldy smelling cup or glass so I prefer cup up.
What you do at home is up to you. I hate a musty/moldy smelling cup or glass so I prefer cup up.