The criterion for 'curious' badge is to post 5 'well received' questions. Although the answer to this is probably quite simple, still, I do know what well-received means in practice.
Qualifying for the curious/inquisitive/socratic badges requires accumulating good asking days. One good asking day means:
asking at least one positively-scored question (aka "well received question") without having another question negatively-scored, closed or deleted on that day.
For curious one needs 5 good asking days; inquisitive 30 and socratic 100. Also, the question record needs to be positive.