For new development
In general, new feature development occurs with support for the lowest Java LTS version covered by Oracle's Premier Support (which typically lasts 5 years from initial General Availability). If the minimum required JVM for a given library is changed, it is accompanied by a semver major release.
Java 11 and (in September 2021) Java 17 are the best choices for new development.
Keeping production systems current
Google tests its client libraries with all current LTS versions covered by Oracle's Extended Support (which typically lasts 8 years from initial General Availability).
Google's client libraries support legacy versions of Java runtimes with long term stable libraries that don't receive feature updates on a best efforts basis as it may not be possible to backport all patches.
Google provides updates on a best efforts basis to apps that continue to use Java 7, though apps might need to upgrade to current versions of the library that supports their JVM.
Where to find specific information
The latest versions and the supported Java versions are identified on
the individual GitHub repository