I am attempting to set up GeoPortal 1.2.4 and use ArcGIS 10.2 as my client desktop. I haven't come across any documentation which says these are compatible other than the Git page. However, I am unable to set up my software, I am getting a message which says 10.1 is not installed.
ArcGIS Desktop 10, 10.1 or 10.2 is required for using the Geoportal Clients for ArcGIS Desktop. The Geoportal Clients are offered in versions specific to your ArcGIS environment; i.e., use the 10.1 Geoportal Clients with ArcGIS 10.1, and the 10 Geoportal Clients with ArcGIS 10. The Geoportal Clients allow users to connect to the geoportal from within ArcGIS Desktop to publish and search. Caution: The ArcGIS Desktop .NET Framework option must be enabled during installation of ArcGIS Desktop on any machine that will be using the Geoportal Clients.
There is a good-looking Esri installation guide published as a white paper: How to Set Up Esri® Geoportal Server 1.2.2 on Windows: Aggregate Geospatial Resources with Open Source Technology.
Esri has published two desktop clients 1.2.4 - one for 10.1 and another for 10.2, so this implies that geoportal is supported with ArcGIS Desktop 10.2 indeed.