Right now, diamond moderators and users with at least 15k reputation can protect questions, though for non-moderators the question must be at least a day old. There's also an auto-protect mechanism for questions with three or more low-rep user deleted answers. Ideas to auto-protect questions with more than N answers haven't been implemented.
It seems that the absence of an entitlement for non-moderators to protect questions less than 24 hours old reflects the idea that one cannot easily judge whether a "fresh" question is in need of protection, and so only moderators should take such action. However, in case of a new question that rapidly attracts answers, assessing the need for protection is easier (if only by perusing the various answers). Therefore, I suggest to allow non-moderators to protect questions less than a day old if the question has attracted at least, say, 10 answers.
(My case in point is As an expert, can you always use TeX for (nearly) any kind of document? which was asked at January 23th at tex.sx. Perusing its first 10 answers, it seemed quite clear to me that the question attracted answers of varying quality by new users. When I was finally able to protect the question, the number of answers had risen to 15, and it is now at 18. 4 of the 15 answers preceding protection have a score of 0 or 1.)