I know I can set the render border with Ctrl+B and dragging a box. But is there a way I can exactly specify the border dimensions?
You can access the render border values via the outliner if you change its display mode to 'Datablocks':
You can then navigate through the tree structure of the outliner to the border values via Scenes> Scene (may be different depending on the name of your actual scene)> Render Data:
The border values can be set from 0-1, so unfortunately you can't set exact pixel coordinates without some maths. 0 is the left side of the camera and 1 is the right side of the camera for the 'x' values (or the bottom and top sides of the camera for the 'y' values).
For more control I've written an Add-on which allows to set the Render Border in Camera View:
Install the Add-on via File > User Preferences > Add-ons. Once it's enabled, use Ctrl+B in the Camera View (Numpad 0), select the Camera and go to Properties > Data > Render Border.
Shifting the box