The beta badge is awarded to those who actively participate in the private beta. This phase occurs directly after the Area51 committment phase. The private beta typically lasts for 7 days, and is open only to members who committed to the Area51 proposal.
The goal of the private beta is to seed the site with content for the public beta. The beta badge is awarded to those who participated during this period.
PMSE's private beta began on February 7th, 2011 and ended on February 14th, 2011. PMSE is currently now in public beta, which means it's open to anyone.
For further reading, check out What is Private Beta Status.