Yes it can absolutely be mandated. If you live in an area with a high crime rate or an area that has a building with street access simply through usage of a fire escaape, then your landlord has an absolute responsibility to minimize danger to you and other tenants. Simply asking or demanding you close your windows (and presumably lock them) when you are not home is reasonable. It would be the same as if you had a ladder propped up against a window on the second story if you lived in a single family residence. The water damage issue is a lesser argument but the risk of entry is enough. The landlord coming in to your home prior to giving you notice regarding this window closure requirement is not appropriate but he or she should put that in writing and indicate to you that entry will occur if you are not home and the window is open. If you are still unsure, contact your state's consumer protection agency that handles landlord tenant matters and see if it can help mediate this matter.