I've seen this banner on many many bounty questions. Some have a pretty healthy 150+ views. Still the banner remains.
What is the criteria for this banner to be put up next to a bounty question? When will it go away?
The criteria for it appearing is that the bounty-offerer chose it as their bounty reason. It's a pretty popular one.
The "question has not received enough attention" message is the reason someone put a bounty on it.
The notice will stay until the bounty is over — i.e., the week is up or it's manually awarded to an answer. When it expires, the notice will stay up for 24 more hours (the grace period), and after that, if the bounty still wasn't awarded, it'll stick around a bit longer, announcing that the grace period has ended.
It will be removed when the bounty has been awarded to an answer, or when the bounty has expired. It is not related to the number of views that a question has garnered, indeed you could apply that bounty reason to the most viewed question on StackOverflow without issue :)
The banner is intentionally put there by the bounty giver--there is no 'criteria'
Basically, when offering a bounty, you have to choose from a generalised reason. One of them is 'Draw attention--this question has not recieved enough attention'. You can choose that regardless of how much attention the post actually has gotten. You could offer a bounty on a 5kviews question with this reasoning, and the system won't complain.
Offering a bounty gives the question attention, and the bounty banner stays till the bounty is awarded or expires(7 days).
Usually, before answering a bounty question, it is good to look at the bounty rationale--since maybe the bounty-offerer is looking for something else. Many times I've seen a bounty question, and thought "Oh, I could answer that", till I see that it's marked "Authoritative reference"?