There is an interesting presentation on the impact of dust on aircraft here. And the Flight Safety foundation has a good article that breaks down a lot of aspects here and is worth the read.
It's not just the reduced visibility, but the associated high winds that can be a real danger for aircraft operating in a dust storm.
Dust storms pose a significant hazard for aviation. Not only do they
drastically reduce visibility, they also are associated with very
strong winds that can seriously affect an aircraft in flight. Engines
can be damaged by ingesting the dust.
But as per your questions directly
How do modern airliners and her crews deal with a large sandstorm?
They most likely will avoid them as they would any heavy weather activity. The extremely reduced visibility might bring the airport below operational flight minimums, thus making an approach or departure impossible.
Would a standard weather briefing warn a pilot of the possibility of a
One main issue with dust storms is how fast they come up, and this can make them harder to predict than a storm that is rolling across a region.
It was a typical early summer evening in southern Arizona, U.S., this
past July 5. At 1751 local time, the Phoenix Sky Harbor International
Airport was reporting 10 mi (16 km) visibility with winds of 7 kt. But
all that changed in a matter of minutes.
As for what to do when you find yourself over or in a dust storm check this article out of a firsthand account.