The awarding of beta badges is triggered manually at some point during the private beta - 37 users now have this badge.
You become eligible for the beta badge once the site moves into public beta, assuming you meet the requirements outlined below:
silver; awarded once
Participate actively in a site's private beta phase
Historically, active participation meant earning at least three bronze badges, but the exact requirements are no longer disclosed to prevent users from doing just the bare minimum.
Not awarded on Meta sites, including Meta Stack Overflow
From List of all badges with full descriptions