I think this really comes down to your preferences as a package maintainer, and how disruptive/how much extra work is required supporting older versions of 1.x Julia (including test run time) becomes for your package as time goes by (e.g. you may want to rely on new language features that make supporting 1.0 difficult or impossible).
I think an entirely pragmatic approach would be to explicitly test only against the latest stable release and whatever 1.x version is the LTS version (this is only my view, primarily as a user of Julia, not a package developer). If the package works on the LTS version, Julia’s adherence to SemVer means that any versions in between the LTS version and the newest stable version should also work (unless you’re relying on undocumented features). Currently the LTS version is 1.0, but (as I understand it) that isn’t set in stone, as outlined by Stefan:
I take Alex’s initial post to mean that 1.1 won’t be a LTS branch, but my reading of Stefan’s comments is that future minor versions (e.g. 1.2) could replace 1.0 as the LTS branch if there is demand for that.