Why is fear the most prevalent emotion in the society of Anthem?
Anthem is one example in a long series of examples of what is usually referred to as "Dystopian Literature"; that is, literature which depicts the antithesis to the usual Utopian concept of a totally free and progressive society which has eradicated all its crime, disorder, etc. and has evolved into a state of social perfection.
Like Brave New World and 1984, Anthem presents Ayn Rand's view of such a society, which appears to be Utopian, but in reality every aspect of the society is corrupt in some way or other. In almost all dystopian stories (with very very few exceptions) society is controlled by some central oligarchy or a network of nodes of control to keep the members of society from deviating from the expected happy bliss of complete compliance with the rules. In Rand's world, a fragment of post-apocalyptic New York City, the people have been thrown back into a medieval world of intense control. However, as Rand points out in her works over and again, man's liberty and individualism is an end in and of itself, so conformity to rules must always be consensual, not forced. Force is the ultimate violation of the individual, so it is the centerpiece of any dystopia. In this world, fear is ubiquitous and firmly implanted in the minds of the characters. They are several generations removed from the war, so over time have even forgotten to question the authority of the Councils. Like sheep, they follow along and don't resist. The councils and heads of homes are the disciplinarians in the society and are effective in keeping their stranglehold on compliance and conformity with several mottos to support the suppression of individualism and inculcating fear in anyone who might dare to resist.
Thus, fear is a constant tool used by the controlling forces and keepers of the societal mechanism of control and production. Only when the protagonist evolves enough to see his own individuality is he able to begin to act as an individual, face his own fears, and then defy his society's laws. His eventual liberation is a Rand new dawn of life free from institutionalized collective life.