The propagandist who called himself Azym Abdullah didn’t need much money to set up a website for ISIS that would broadcast gruesome beheading videos. What he needed was secrecy, so in 2014 he reportedly turned to cryptocurrency.

He paid a little more than 1 bitcoin, approximately $400 at the time, to register the domain name in Iceland and host it on servers around the globe. His site asked visitors for donations to help pay for the upkeep. Those, too, were in bitcoin.

Sending donations that way allowed his donors to...