Users with more than 15k reputation can protect questions. I would argue that it doesn't make any sense anymore to keep this privilege since the automatic protection of questions with three deleted answers by new users has been implemented.
Protecting a question has a practically invisible, but still significant drawback: It can prevent good answers from ever being posted by new users. I think that protection is somewhat overused, and that we, the moderators and the community, should be more careful when to apply it.
Shog9 stated in no uncertain terms that
proactively protecting questions is a stupid idea, and you should never do it even if the system does allow you to do it.
I agree with that, and my personal guideline on when to protect a question fits very well to the automatic protection mechanism, which is after three removed posts from new users.
There might be some exceptions, but in general I would say that we just don't need to protect questions earlier than that. I almost never use my moderator or 15k power to protect questions and would not miss it at all.
I think that the exceptions can be handled by the moderators, though even those should probably try to avoid protecting questions if there isn't a clear record of bad non-answers from new users on that question. There is no need for this as a regular user privilege and it should be removed entirely, or replaced by some other privilege.