"Are we going to worry about the cost benefits of building a system to stop an asteroid from hitting the earth?"

Of course.
It's an absolute fact, that there is an asteriod which is presently "on course" to hit earth. It's even a safe bet to say an asteriod will impact earth within a couple a months. The question is when & how how big.
There are about 30 impactor hitting Earth every year with the explosive power of a lot of dynamite or small nuclear bomb. I have never heard of anyone bringing it up, but the small impactors from space (from a grain of sand to something the size of a car) are probably adding to global temperature. Is this a significant amount? No, probably not. Would it be a measurable amount as compared to human activity on global temperature. It's possible.
I couldn't rule this out without some more information- the total effect could instead be even be cooling the planet (slightly). Is it possible to stop all impactors from hitting Earth, probably, if you want to spend tens of trillion of dollar you could stop at least most of the mass of impactors from hitting our atmosphere- but it wouldn't be done, because it's not cost effective in terms of lowering global temperature (or even saving lives- not at trillions per year).

"If we can prove that it is necessary, and we can do something about global warming, I would think we should be willing to spend quite a bit to stop it."
How much would you be willing to spend per year?

"Saving the world from a global disaster should be cost effective whatever the coast."
(Well, personally I would like to save the Pacific coast from an impactor hitting it without me having some advanced knowledge, but that's another story.) I assume you meant cost:) A mile in diameter rock hitting anywhere on Earth would fit my definition of world disaster. Whereas the global temperature rising (or lowering) by 1/2 degree over 10 years does not fit my definition of world disaster.
Nevertheless if we spent the same amount on studing global warming as it did on detecting asteriods, I could live with that.