I am rendering a image in cycles.The question is how to blur background in blender cycles?
There are many ways of doing this, here is how I would do it.
First, go to the render layers tab of the properties view and enable an environment pass under the passes rollout.
Then make sure you have checked transparent under render properties > film rollout.
Then you can blur just the environment pass in the compositor and mix it into the render using a mix node based on the alpha.