The Red Keep is the seat of the current ruling King, and is also the location of the Iron Throne. Located in King's Landing, the King and his family, as well as his Small Council, can be found here.
The Red Keep has been the seat of the head of House Targaryen since Aegon's Conquest, while the Crown Prince or Princess has their seat at the ancestral seat of House Targaryen: Dragonstone.
Since Robert's Rebellion, Robert Baratheon has taken up residence with his family in the Red Keep. His concern about the White Walkers is minimal, and is in no hurry to rally King's Landing to try and find the Dragonglass weapons of either houses Baratheon or Lannister.