This is a red herring: even if Apophis hits (current estimates 1-in-many-thousands for 2036), it's tiny, too small to be an existential risk (see wikipedia quote below), to a first and second approximation. Asteroid risk comes from the chance of huge impacts, the 1 in 100 million years variety. Spending several million dollars on tracking asteroids (we have found 90%+ of such big asteroids) was a great use of money by most standards, but you get diminishing returns as you move along. From an x-risk point of view, my take is that we should continue surveillance, and in the very unlikely event of spotting a huge asteroid on a dangerous path we should obviously mobilize much of our civilization to deflecting it.

From a public health point of view, concerned only with current people and valuing rich country folk at hundreds or thousands of times the value of poor-country folk, much larger expenditures are warranted, but that's not an existential risk issue, by and large.

NASA initially estimated the energy that Apophis would have released if it struck Earth as the equivalent of 1480 megatons of TNT. A later, more refined NASA estimate was 880 megatons, then revised to 510 megatons.[3] The impacts which created the Barringer Crater or the Tunguska event are estimated to be in the 3–10 megaton range.[25] The 1883 eruption of Krakatoa was the equivalent of roughly 200 megatons and the biggest hydrogen bomb ever exploded, the Tsar Bomba, was around 50 megatons. In comparison, the Chicxulub impact, believed by many to be a significant factor in the extinction of the dinosaurs, has been estimated to have released about as much energy as 100,000,000 megatons (100 Teratons).

Waiting until after the "2012 prediction" is behind us would be best. It seems most people can't incorporate both the Mayan End of Days and climate change into their future plans and goals, so adding a third 'hoaxable' problem to Earth's future won't help.

With that said, I can see society accepting a killer asteroid over climate change - since the science for the latter is more abstract than "hey look at this asteroid coming right at us."

End of the World movies and books are overdone, so there wouldn't be a need to break the ice to the public through popular media. I think it would be safe to simply state the danger directly.

It seems fear would drive some people to insane things, but enough people to be driven to help the situation, ie. agreeing to help fund, through taxes, research and operation of a mission to divert a killer asteroid.

On an outside view asteroids seem like an astronomically (har har har) unlikely way to go, since the Earth's been around a long time and has only received a few extinction-impact asteroids. (Anthropic considerations should discount the likelihood of asteroids that would wipe out all life on Earth, but not ones that just rearrange it like KT.)