In the case of ANWR, the very best estimates are a savings to consumers of about 3 cents per gallon starting in about ten years.

The argument for ANWR among its proponents is not about gas prices at all, as the prices wouldn't be affected in any practical sense, but about independence from foreign oil. That would be a boon to the industry since they would own a vertical monopoly of production, of no real benefit to consumers, and would have a impact on the environment.

Well I am against drilling in ANWR and even Oil Dependency but to be fair it would be more of a long term thing for gas prices since alot of the initial costs would be the set up costs to build the drilling rigs and even moving personal.
Oil in itself has more purposes than just gasoline.

I do believe one day we will have to drill there but not for at least another 20-30 years and I would consider it a wiser investment to start moving towards electric cars and improving battery technology as well as focusing on improving solar and IR energy panels. An interesting technology would have been the thorium based independent nuclear reactors but that probably scares too many people.