Your Fleet's Starbase can give you a variety of small advantages in many of your activities as a Captain. Sometimes, however, it might not be apparent that those advantages are present.
The Bonuses provided by a Starbase are automatically calculated into the values that you are shown during Away Missions, Cadet Challenges, Shuttle Missions, and Voyages, as well as your Chroniton maximum shown on the main interface.
The best way to determine which Bonuses you are currently benefiting from is to go to the Starbase section of your Fleet, and tap the button at the top-left that reads "Bonuses". There you will find a list of active Bonuses being applied to your actions, and if any of them have changed since you last looked, they will be marked as "New".