I am running ArcGIS Server 10.4.1 and have installed PostgreSQL 9.6. However, the latest version of st_geometry.dll listed in my installation directory is for PostgreSQL 9.4. Is there a 9.6 version available, or is this an unsupported combination?
PostgreSQL 9.6.x is not supported with ArcGIS 10.4.x or even 10.5.x.
Minimum supported database versions
The following versions of PostgreSQL and PostGIS are supported with ArcGIS. The specific versions listed are the minimum minor version supported and certified for each supported major release of PostgreSQL. Newer minor versions are supported, but are not certified.
PostgreSQL 9.4.5 (64 bit)
PostgreSQL 9.3.10 (64 bit)
PostgreSQL 9.2.14 (64 bit)
I think you meant PG 9.6.x is not supported by Esri for enterprise geodatabases, yet.
It still might be possible to access a non-geodatabase instance of 9.6 with PostGIS 2.2 (though I haven't tried). You won't be using SDE.ST_Geometry with PG 9.6 until at least ArcGIS 10.6, since extension DLLs are locked to a specific PG release (9.3.x, 9.4.x,...), and 9.6 is not supported.