As long as the crosswind was within company limits just before touchdown, there is no other wind limit that applies apart from the problem parking. High windspeeds can turn aeroplanes when parked, which can cause big trouble if attached to a towbar or on a jetty, so there is usually a limit on operations on the ground. At Leeds this summer, we unloaded the passengers on the jetty, but there was a sudden panic to get the aeroplane towed away and into wind in case it caused damage when swinging. Years ago we used to park a HS748 at Sumburgh, Shetland Islands. Some mornings I used to wonder whether it would still be there!
These limits vary with type, so airport operators do not close the airport- they leave the decision to pilots.