The most convenient way would be a browser console script that goes through the 2400+ animations and adds them to your assets. Of course that is only reasonable as a backup of sorts (say, Mixamo drops the service with no prior notice …as they did with the UE4 format), otherwise the best way is to work with the site’s search and customization options.
It just so happens I’m working on such a script right now.
In the first pass it goes through all animations and adds them to your assets,
then in a second pass it creates packs, 75 anims each and auto-sets checkmarks for “mirror” and / or “in place” according to your preferences.
It basically remedies the site’s lack of multi selection options.
All you have to do then, is go into “My Assets” and download the packs.
Hope such a script doesn’t violate any ToS, but I can’t imagine how it could.