I tried to search here on meta, but I couldn't find any thread that answers this question.
What are the criteria for awarding a beta badge on the stackexchange sites?
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For the SE 2.0 sites the requirement is:
Actively participated in the private beta
So this means
You must have committed to the site so you can gain access to the private beta.
You must make a minimum number of posts. I think it's 10, but I can't find the reference right now (it's on one of the new sites meta). However, Gamecat has a beta badge with only 3 answers.
There might be a minimum number of votes, but I have no evidence of this.
The beta badge appears to be awarded with the your Area 51 commitment for that site being "fulfilled", but as Gamecat points out they aren't related.