How to make a white boarder inside a black frame.

In general, the process would be to stack strokes using the Appearance panel. For instance, say you made a rectangle with a light blue fill and a 4 pt. black stroke. To add a 4 pt. white stroke inside that black stroke, you would do the following:

• Open the Appearance panel (Window > Appearance)
• Click the Add New Stroke button on the bottom left of the Appearance panel which creates a new stroke of the same color and weight as the original
• Select the original stroke (or move the new one down--really you just want it to below the stroke that you'll leave as the black one)
• Change the color to white (you'll notice that it doesn't show up--that's because it's the same weight as the other)
• Change the stroke weight to 8 pt (or however wide you want it with 4 pts of the black stroke obscuring it)
• If you want the white stroke to appear inside of the black stroke, click on Stroke for the white stroke and change Align Stroke to Align Stroke to Inside

You can also have the white stroke on top in the Appearance panel if you set the alignment of the black stroke to outside and the white stroke to inside.