Oil is a global fungible commodity. Any additional oil reduces the tightness of supply and reduces concerns over possible shortages.
But the US lower 48 is the closest oil importing market. Sending the oil to other locations would reduce its profit with increased transportation costs. The vast majority of that would stay in the US.
In the end ANWR would have little impact at the pump.
Most importantly, if ANWR did go online the OPEC "Hawks" would cut production thereby maintaining the current price.
Drilling ANWR is a great idea but it is not the answer to cheap oil.