How did the Cold War affect the United States goverment?

First, look at how the Cold War affected the U.S. government in terms of foreign policy.  Due to a threat of nuclear war with the Soviet Union, America had to always have nuclear preparedness plans with the capacity to deliver these weapons, whether through bombers, missiles, or naval delivery systems.  This was money that could have been spent elsewhere, such as on domestic programs.  Also, America got itself into foreign entanglements such as Vietnam and Korea in order to contain communism, killing thousands of young men and injuring many more.  America also got itself involved in covert actions in Guatemala, Iran, and Cuba in order to support rightist, U.S. friendly governments.  Many in the developing world still do not trust that America is the beacon of democracy that it claims to be, but rather they think that America only wants leaders that will act in its interest.  

Domestically, the Cold War ensured that there would be a draft for the military well after WWII ended.  America encouraged more people to go into science and math fields in order to ensure it would have more scientists than the Soviet Union.  The FBI kept reports on people who it suspected may be Communist--this is even more controversial today as more information comes out about the program.  The Interstate Highway system was designed by Eisenhower to be used in war, but it really helps Americans move from place to place easier and has promoted the growth of suburbs at the expense of the inner city.  While it never turned into a true "shooting" war, it is hard to imagine America without the Cold War.