At the international space port airport I worked at, the Aerodrome Duty Manager would do a runway inspection and decide. That said, if ATC thought it was potentially dangerous, they would request that inspection to happen.
Thus; ATC would normally start the action toward a closure be it by supplying the weather observation or other information, but the aerodrome authority is 'responsible' for the actual closure.
THAT SAID:
At the two regional airports I worked at, ATC were the designated authority on the airfield. Basically, we have approval to make the decision on their behalf. ATC will go out on a runway inspection with ops & decide based on the books / professional knowledge / experience whether its safe enough to use. You have to be able to justify it, but isnt that the way of the world today?
