There is no benefit to anything (planets or ships) within a resource mining starbase area of influence.
As pndrev says resource mining starbases give you a bonus to your global abilities. Pndrev mentions morale, economy, research, influence resources which give a bonus to the appropriate ability. These are listed in your Stats & Graphs area of your civilization manager. Also military resources give a bonus to both weapons and defenses. Each fully populated resource mining SB adds a 39% bonus.
Here's a screenshot of the stats & graphs page from a gigantic game that I'm playing. Note the circled areas are effected by the resources I'm currently mining. In this game I have the following resources fully mined giving the following benefits. Anything above these bonus comes from racial abilities, tech or wonders.
6 economics = 234%
5 military = 195%
4 research = 156%
8 morale = 312%
6 influence = 234%

If I didn't have these resource mining SB's then I would have the following total bonuses.
Economics = 171%
Weapons = 6%
Defense = 14%
Morale = 120%
Research = 42%
Influence = 74%
See what a difference these make.