Hello,
what is your problem exactly. Do you use Safari on Windows or Mac?
I have tested both.
Safari on Windows has no problems with downloading worlds on this site.
Safari on MAC has some problems with downloading worlds: Server not found, downloading stopped, downloading started. This three notifactions i get by download a world with Safari on Mac.
My tested Solution: (on my Macbook Pro)
Performing all available updates, restart system, search available updates until there are no more Updates. My Mac has become many updates because it was about a year off.
I'm from germany, my english is not the best. Speaking for me is easier than writting.