Safari 12 does not recognize Adobe Air

Most of the modern browsers block plug-ins in general, and even Flash Player is heavily restricted these days.  My sense here is that the page is using Flash Player to detect the presence of AIR, and Safari restricts Flash Player's access to the filesystem such that we can no longer successfully make that check.  There's no working around that from the Flash Player side.

More importantly, as we head into the endgame for Flash Player support in general, Flash Player is going to be more and more difficult to run.  If your organization is not currently executing on a migration strategy for your internal applications, you probably should.  In the short term, that probably means removing that check for the AIR runtime.  In the not-too-much-shorter term, that probably means migrating to native web technologies like HTML5 and JavaScript.

Because the browsers are imposing these restrictions on Flash Player, there's not really anything we can do to effect a workaround for you.  You're basically looking at either providing some virtualized mechanism that allows you to access these tools from a software and browser configuration "frozen in time" through virtualization technologies like Citrix or VMWare, or (probably better), it's time to start migrating.  While Flash Player will continue to be supported through the end of 2020, the actual timeline for the browsers to deprecate Flash Player is already well underway, and running Flash will only become more painful as we continue to steward the technology towards end-of-life.

If your organization is not familiar, you can find our official statement, along with the timelines and statements from the major US browser in the article below:

Flash & The Future of Interactive Content | Adobe Blog

What is the actual link you are attempting to go to?  The one you posted is the a redirect for alternate content when Flash isn't installed, or the browser isn't able to detect the content requires Flash (in this case, this is a content issue, not Flash Player issue).  To attempt to reproduce, I'll need the actual link you're trying to get to, not the redirect link.