Press the Gem icon with the plus sign at the top of the screen to bring up the, "Buy Gems" window.  You'll see your Castle, or Base, icon in the upper left.  Press that, and then it will bring up your options.  

You can also navigate to this selection tool from the rewards window in the Event Base.

Quest and Milestone rewards use whichever Base you've selected as your destination, unless there is a Base-specific reward.