The fact that chat privileges are granted early (20 rep) means that many users who have not had any substantial amount of SO/SE experience wind up in chat. The chat room I frequent has had no less than three users ask questions that belonged on the main site in the past twelve hours. As chat is the primary refuge of experienced users, this causes a large amount of consternation.
A few options that I don't see as working or that cause more problems than they cure:
Raise the reputation required to use chat
I'm not very much in favor of this. Early use of chat was key to my deeper involvement in the site, and provided me with quite a bit of insight into how SO works.
This has been discussed, and I generally agree it provides the wrong incentives.
Use existing moderator tools
Seems pretty harsh to moderate someone for a simple mistake given the limited moderator tools available. And still requires too much intervention by experienced users when chat gets flooded with questions.
Here's what I think might work:
Provide guidance on appropriate use of chat
In many other situations, SE provides guidance on appropriate behavior, yet at 20 rep a user can join a chat with no other information about what chat is for than the chatroom description. Either a large warning box on the top of chat.SO or a pop-up the first few times a new user types in the chat window would help. The box could say something like, "Chat is a place to converse about life and discuss technical issues. It is not a place for Q&A. If you have a specific problem to solve, chances are it belongs on the main site as a question, not in chat. This also helps get you better answers, as it is a much better format for passing the details of code."
Here's the guidance that currently exists: A drop-down overlay which points you towards the FAQ. The FAQ itself says nothing to discourage Q&A--in fact, it even implies that it's appropriate.
I don't think that Q&A never belongs in chat, but I have yet to see someone who is not a regular chat user ask a chat-appropriate Q&A.