How to add a CSS class to fields in Drupal 7? I want to add a CSS class to the HTML output of each field in my field collection.

Please recommend me some module or method to achieve this. I have tried the Fences and Field Formatter Class modules but they did not work.
You could use hook_preprocess_field() in which you can define additional classes for your fields.

Check these to get some idea:
http://atendesigngroup.com/blog/adding-css-classes-fields-drupal
https://gist.github.com/1184302