From Admin>Advanced>System Logs in the Artifactory GUI on an HA cluster, there is a single "Download" button.
Pressing the download button will download logs from one of the HA servers. There is no indication as to which one, though.
The "Download" button (or the subsequent dialog) should indicate (or even better, allow the user to select) from which HA node the download will run.
The workaround is straightforward. Rather than using the Artifactory GUI, inspect the artifactory.log on the desired machine.