This is why it's a known mutagen with a long list of very specific, strident toxicity warnings. That's the organic cation, though; compared to that, the Br- anion is utterly benign. You're looking at the wrong hazard!
Br- is a standard counterion in human-approved drugs (those formulated as HBr salts).
https://www.google.com/search?q=hydrobromide+site:drugbank.c...
Sea salt should have a fair amount of bromide in it. Bromine can be produced commercially by bubbling chlorine gas through sea water.
I'll nip this in the bud and say "DNA particles" was definitely the wrong word.
You are correct, the Bromine here isn't really the problem; the fact that any intercalculation with DNA occurs is the main problem. Is it Bromine's fault? Not particularly, but it's an accessory in this instance. Based on anecdotal evidence it's common to the electrophoresis process.
My reason for bringing this up is just one of those relatable stories (/s) about why I also don't like working with Bromine (even if contextually it's not Bromine's fault, we love you Bromine!)
I assume you meant intercalation[1], right?
[1]: https://en.wikipedia.org/wiki/Intercalation_(biochemistry)