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.
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.
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.