Some time back Doc asked How can we encourage people to read the FAQ before asking their first question? which is a really good question and a fairly consistent challenge to get people to ask questions on the site they should be (we get a significant number of debugging questions for example).
So, here's the suggestion (that Robert Harvey inspired) put into a post.
That "Welcome (back) to the site" banner that shows up when you join a new site? That one that you can dismiss with a click.
Have a link to help/on-topic if the user visits the site and has all of the following:
- less than two up voted questions
- less than 100 reputation on site (akin to how protection does it with 10 rep)
- lacks the "informed" badge (the one you get by reading the tour)
So, a link at the top of the page. Point them clearly and succinctly to the help center.
I have no illusions that this will prevent all the off topic questions. However, even a slight reduction in the "this is the completely wrong site for this question" influx of new questions that we get can help reduce the initial perception of going into a new question of "this question is probably off topic."