As an ATCO, I do not have the authority to close an airfield/airport due to bad weather; as the previous poster has said, this is the responsibility of those who get paid much more than I do. Closing a runway is a similar matter; if the airfield only has a single runway then the ATCO cannot close this; more than one runway and the ATCO can choose the most suitable runway for operating from.
If the weather gets that bad, all we can do is ensure that the ATIS is correct and up to date, and offer what assistance we can to those who decide that they wish to land/depart.