It would ease US-Russia tensions a lot and Russian tensions with Europe. It would be great for the American economy with quick trade, lowering transport costs. It would create more jobs, benefit the economies of both countries in the long and short term while lower Russia tensions could even lead to less military spending in the US, Russia and Europe. Alaska would also be developed well. An all around great idea.
This way everybody wins. Ukraine and Georgia's sovereignty is secured. And then we can lift the sanctions. In exchange we pay to fund the Bering Strait Crossing. This would be a boon to the US and Russian economies in particular. But also a boon to the world economy, since it would be easier to transport goods and tourism between the American supercontinent and Eurasia.
This would also boost the growth of Nome, which currently isn't even connected to the highway system but would almost certainly get a highway connection if we were building a tunnel under the Bering Strait. Development would also be possible on the Diomede Islands in the middle of the strait. Naukan, currently just a small village in Russia would need to be developed into a major city, which would bring development and economic growth to Russia's Far East and consequently to Russia as a whole.
Part of Russia's behavior in foreign policy is the difficulties in managing resources, particularly in the cold tundra of Siberia. By connecting Russia to United States and increasing trade on both sides of the Bering Strait growth would be encouraged in Siberia as well, increasing economic security in Russia and leading to better relations.