I am married to a Caltech grad. I assure you, it is very selective and difficult to get into Caltech as an undergrad. My husband's freshman class had only 210 students in it, and not everyone managed to complete their degrees there. He did. Sort of like Sheldon and the others, he didn't really feel like he fit in until he went to Caltech and met more people like himself. You most definitely need to be a very serious math and science student to get into and survive at Caltech. My husband is brilliant and very, very smart, but fortunately has way better social skills than Sheldon does.
As for this show making Caltech more known to the average American, my husband used to travel frequently for work, and often wore a Caltech sweatshirt while traveling. A TSA agent noted his shirt and said, "Caltech. That's the fictional school from the Big Bang Theory, right?"