The first version ─ the one with only one em-dash ─ is correct.
When an author uses two em-dashes within a sentence this way, the purpose is to fully section off a piece of text to create an interruption in the sentence, one that is typically in some way independent or even separable from the sentence. Here's an example of this use of two em-dashes:
When Besso left Zurich, he took the manuscript with him ─ which is
viewed as the reason for its survival ─ and took a job in the patent
office in Bern.
The information within the dashes here makes a point that, while related to the sentence, is nevertheless independent from it. Indeed this segment can be removed from the sentence and the remaining words will still make sense. However, to take your text (the version with the two dashes):
When Besso left Zurich, he took the manuscript with him ─ which is
viewed as the reason for its survival ─ as Christie’s said Einstein
probably wouldn’t have bothered to “keep what he saw as a working
document.”
The second em-dash is at best stylistically weak here, and at worst confusing and wrong, because it creates a "middle section" that the reader expects to be an independent point. But this middle section is not an independent point, but is rather essentially linked to the right-hand section; i.e. the piece of information on the right side of the second dash (describing Einstein's habit of throwing away working documents) explains what's between the two em-dashes (that Basso's taking it is the reason for its survival). Removing the middle section would yield nonsense.
A single dash within a sentence, on the other hand, doesn't really have the same effect of creating an independent item within the sentence. Rather, it's a stylistic device to break up a sentence, and can be used when something more emphatic than a comma is desired, as is the case in the first version of the excerpt. A comma, however, would be perfectly fine here as well.
