The problem with trying to pass laws in order to "feel safe" is that there has never, and will never be, a law that a smart enough criminal can't get around. The only people who suffer under restrictive laws or privacy invading laws are we, the people who have no ill intentions to begin with.
So far as the idea of getting a background check card goes, what real difference would it ever make? After all, the hijackers created their own passports, that while not absolutely 100% perfect, were perfect enough that nobody stopped them or said anything about it. If they do go on to make a background check card, it will also be forgeable and what have you really accomplished except giving yourself that warm and cozy false feeling of safety, while also giving up very much of your private life to be cataloged and used by the government as they see fit. Doesn't seem like a very good trade-off to me.
The bottom line in my opinion is basically what Bravo25 said, if you are expecting anyone else to provide for your safety other than yourself then you are only fooling yourself into a false sense of security because nobody is better motivated nor in a better position to see to your own safety than you.
I will guarantee that if I'm ever on a hijacked plane that I won't just sit back and wait for someone from government law enforcement to come and rescue me like a helpless infant. I will fight my own battle just as anyone would have back in the days before everyone in this country became weak and dependant on the government for their security. I keep a loaded gun within easy reach whenever possible and I don't rely on anyone for my own security except for myself, and ultimately God.
I also don't choose to live my life in fear from any crazy Arab Muslims or any other boogy-man the power mongers in our government may hold up and try to frighten us into submission with. The only entity that I truly do fear, and is impossible for me to fight alone, is our own government, which if not kept in check will abuse it's power over us every chance it gets, or I should better say, those wicked people within our government will abuse whatever power they can obtain over us, and *knowledge is power*. This encroachment of government into my private life has been, and will likely continue to be, the biggest threat to my own personal security, barring a full scale war with another equally powerful nation of course.