If you're using a Wand tool to select portions to cut out, try turning the feathering down, and doing "Expand selection" by 1 or 2.
If you're doing the erasing by hand (eraser tool), try turning the hardness up to about 85% or so.
Either way, It's best to make a pure black bottom layer when you're doing this stuff; so you can actually see any white lines that might occur.
This really is pretty bitchin'.
What was the border originally a picture of?