Modems are still used extensively for “out-of-band” communication, such as when a high-speed link to a remote datacenter fails; the telephone network is often more robust than broadband, even in the business world. Also, modems are usually hooked directly to servers, bypassing firewalls and other security controls.
Broadband connections use modems as well...most any connection I can think of does. DSL modems, cable modems,....