Nobody can predict what will happen between the U.S. and Russia, but I'd be really surprised if things got so bad that U.S. companies didn't feel comfortable shipping goods through Russia. It's not like we're talking about a third-world country or anything.
And what you say about damage is downright silly, because the same concern applies equally for a bridge inside our borders. In fact, by your standards, the docks where those boats load their cargo should never have been built, because if one of the minimum-wage immigrants carrying cargo on his shoulders out to a small boat in waist-deep water dies of a heart attack, it doesn't prevent other workers from loading cargo, whereas if a dock collapses, it does, and those workers can be used for other things if we suddenly no longer need boat shipping. I mean, the only way that logic even starts to make sense is if a serious failure is highly probable, and if that's the case, then it means they got the design wrong.
Besides, the cost of a Bering Strait bridge could be a lot lower than you might think. They would need one segment of it to be tall enough to let shipping traffic through—possibly between the two Diomede Islands—but the rest of it could ostensibly be a simple pontoon bridge, which is relatively cheap.
Most of the cost of the project would likely be for that one span between the two islands that's tall enough to let ships pass under it. That would cost several billion dollars, in all likelihood. The remaining 55 miles, assuming other pontoon bridges are any indication of cost, should be the neighborhood of $5 million to $10 million per lane-mile. At 55 miles long, a four-lane pontoon bridge should cost a couple of billion dollars, give or take, which is about as much money as we waste on a single B-2 bomber.
Of course, a pontoon bridge in that area would have to be specifically designed to withstand the rather severe storms that the Bering sea experiences, which could drive the cost way up. On the other hand, the project is so huge that economies of scale would kick in and bring the component cost way, way down (because you'd be building over 18,000 identical 16-foot segments), which would probably balance that out to a large extent.
Of course, I am not a bridge engineer, so my estimates could be way off, but I wouldn't be at all surprised if someone were able to come up with a design that fell under the $10 billion mark, or about twice the cost of the Bay Bridge. Heck, the tunnel that Russia proposed was only sixty or seventy billion, so that estimate probably isn't too far off the mark.