There should be an option to downvote in chat.
Chat is very tied to SO, and almost all rooms have well defined basic rules.
I believe strongly that being able to downvote off-topic or run-by questions in chat would very much discourage such irritating behavior. It is also much more humane than letting room regulars either completely ignore or pound the new person in to the ground.
(edit) The current system is lacking in user-self-regulation like on all the other SE sites. Room owners aren't always as present and vigilant as downvote-capable users (that are active and present at the time of the incident) could be.
I speak out of my personal experience (it's not me being ignored or drive-by-question dropping, it's others) in the Lounge, which is a great room if you understand the relaxed mentality governing the regulars' chatting behavior.
EDIT To respond to the comment of "the downvoting system is not described here". This is a discussion post, meant to discuss the option. The implementation depends on the outcome of this discussion, obviously. I was thinking
-5 rep on the "mother site" to which the user's profile is linked. This is just an example!