Piping as Julia suggested will probably produce the prettiest checked border effect.
I'm supposing that you're working with a square or rectangular cookie.
Airbrush might work, but either way, you'd need to measure the perimeter of the cookie from corner to corner, to determine how the checked pattern will fit in order to get a consistently accurate placement of checks. Does that make sense?
There are some cute picnic cookies on pintrest with tiny ants, the bane of most picnics, on an airbrushed red and white checkered tablecloth. But of course the entire cookie surface is checked.
Sweetambs has recently put out a tutorial for a plaid cookie, again, the entire cookie is plaid. Her technique may be helpful to you though
Please show us how the cookies turn out! I bet they'll be great.