Enable new modules dependencies on deploy
There is a feature request issue in a Drush queue, just stumbled on it. In donquixote's Drux project are commands you need:


  drux-enable-dependencies (dep-en) - Download and enable dependencies of existing modules.
  drux-find-obsolete (obs) - Find modules that are not direct or indirect dependencies of the modules given as arguments 
  drux-generate (obs-gen) - Generate a list of dependencies[] = .. to copy+paste into a module info file.
  drux-list-dependencies (dep) - List dependencies that would be enabled with drush dep-en. (The help text seems to be wrong).


So for the task at hand, drush dep-en should do the job.

You can download it here.

UPDATE:
Now available on github: https://github.com/donquixote/drupal-drux

Alternatives, part I

As pointed out by Jesse Pinho in another answer, you can also run drush fr DEPENDENT_MODULE.

Or you can run drush en DEPENDENT_MODULE.

The difference is that for these commands you need to know the "DEPENDENT_MODULE" which has the added dependencies.

With drush dep-en you can just run the command without any arguments, and it will check all enabled modules for dependencies.

Alternatives, part II

Instead of such drush commands in combination with dependencies, you could also use hook_update_N() to enable additional modules. This seems a common technique for Drupal 7 projects.

See the answer by reubidium.