"Similar to the citizens of Russia during two world wars, Americans have now been branded with the memories of a traumatic, unremitting long term catastrophe." Should it be 'branded with the memories....' or 'branded by the memories...' Which one is proper?
Either would apply, but have different meanings. To the degree that I can guess your intentions, 'with' is probably the better choice. However, the metaphor itself makes little sense, because a brand is a mark visible to others. Memories are not a visible mark. That's probably part of why you're so troubled by the sentence.
You might try changing it to read "... Americans have now been burdened with the memories ..." That avoids Cogito's objection, and it scans the same way.
It makes better sense but it doesn't communicate the emphasis he wants to give the sentence. Maybe "..ridden with the memories..." would be more appropriate.
Traumatised people very often feel like they've been branded, like everyone around them can see that they've been "marked" with such a uniquely devastating experience. It is as common in rape survivors as it is in war veterans and major disaster survivors. I think it's an apt description and "branded with" sounds right to me. With metaphors and word choices, I find that ultimately the writer has the wonderful opportunity to enhance collective language. If you take a sentence out of context, some people might think that "branded" is the wrong word. But if through your writing, you make them see why "branded"is the right word, they will be enriched as readers.