I just checked Logitech's page and this is their current support:
As of macOS 10.15 Catalina, 32-bit software such as MyHarmony and Logitech Harmony Remote Software will no longer run and will require an update for it to work. If you’re using a hub-based remote, you will need to use the Harmony App on your iOS or Android device. You can also continue to use MyHarmony on a computer running Windows or macOS 10.14 or lower. If you have a remote that doesn't use a hub, you many download the latest software using the links provided on this page.
With the release of macOS 10.15 Catalina, 32-bit apps such as MyHarmony will no longer launch and instead display an error message. For more information, see Apple's announcement regarding their transition to 64-bit technology.
Took a while to dig up a page where they mention support for Catalina. But it depends on which model you have:
If you have a remote that doesn't use a hub on MyHarmony, our new 64-bit app for Catalina, Harmony Desktop, is available for download. Harmony Desktop will only support remotes that do not have a hub; including Harmony 200, 300, 350, 650, 665, Touch, Ultimate One, and 950.
The link in the previous paragraph takes you here. Click on the image at the lower right (Using Harmony without a hub?). A popup will appear with a DOWNLOAD HARMONY button. This will download 64 bit software for a Harmony controller. But as the text above says, it's only for the listed models that do not use a hub.