I saw a bunch of articles about how to integrate Struts 1.3.x with Spring 3.0.x and they all reference deprecated classes.
What is the right and up-to-date approach to do this?
The deprecated classes are the approach if you must implement Struts 1.3 and Spring 3.0.
Struts 1 is quite old and there's little reason for newer Spring versions to continue supporting it.
In other words, if you continue to use old Struts, and old Spring, everything is fine.
If you want to keep versioning Spring, however, you'll run into issues, because the deprecated classes/methods may or may not exist in future revisions.
Ideally you'd migrate to Struts 2 or Spring MVC, but obviously that isn't always possible.