Almost every large airport will have a ontingency plan during adverse weather such as fog, snow and weather etc.
The plan is that airlines will have to cut X% of their movements between a said time in advance.
This would take into consideration the reduction in the flow of departing/arriving aircraft and any other factors.
This will ensure the safe operation of aircraft around the airfield based on the conditions. And in the case of snow it would provide the opportunity for snow clearing vehicles to operate during ground movements.
It is a case of damage limitation i'm affraid and we all seen the complaints when an airport closes especially the larger ones. The airports did not close however some flight disruption is always going to happen during these conditions especially when the airport is almost running at capacity.