Apple commenced its plan to begin phasing out 32-bit apps on Macs in macOS High Sierra 10.13.4. When a 32-bit app is opened in High Sierra, users get a warning about its future incompatibility with the macOS operating system.
Likewise, when opening 32-bit apps in macOS 10.14 Mojave (beta 1), users are shown a dialog box with a similar message telling them that "This app will not work with future versions of macOS". Clicking "OK" on the prompt then allows the app to open.
Currently, the warning is only shown one time for each app. That could well change in subsequent betas of macOS Mojave, however, since Apple previously said it would include "aggressive" warnings about 32-bit apps in the next version of macOS after High Sierra before they are phased out entirely.
Apple's effort to phase them out on Macs mirrors the path it took when ending 32-bit app support on iOS devices. In iOS 10, Apple provided increasingly more insistent warnings to let users know that their apps wouldn't work with future versions of iOS before phasing out 32-bit support entirely in iOS 11.
Once 32-bit apps are phased out on Macs, they won't be able to be used at all, so users will need to find replacements for older 32-bit apps that aren't likely to be updated to 64-bit. You can find out which apps on your Mac are still running in 32-bit by following our how-to guide.