For the people of Cocoon, Pulse is a frightening place. Those who come into contact with it are believed to be elements of a plot to invade Cocoon and are banished from the protections of the Cocoon realm by the "holy" government and its army. The original E3 trailer depicted a group of people who seemed to be confined on a train. These people had come into contact with Pulse and were being quarantined.
Cocoon is protected by crystals. This may not necessarily be the case for the rest of the games in the Fabula Nova Crystalis series, but the crystals in FFXIII are more than just shiny stones with magic abilities. They have the ability to create real life objects. They make life easy for people in Cocoon by creating machines and living creatures.
The relationship between Cocoon and Pulse can be thought of as this, according to director Motome Toriyama. First, there was Pulse, a dangerous world where people reside under the fear of monster attack. Cocoon was made when the crystals considered what would be an appropriate living space for people forced to reside in the world of Pulse.