I'm fairly sure all starbases generate a base level of influence- borders will warp and distort when any base is built, and it doesn't seem to matter much, if any, what type of base it is.  Influence starbases are simply the only ones that can boost that initial influence higher, though none of them can work outside the starbase radius.  It's a fairly minor bonus, and almost nothing compared to the extended range the base gives you.

  The sensor radius is also irrelevent in most cases- an AWACS ship design can do the same job (just design in a whole bunch of sensors), use less money to to it (one ship vs the five Constructors, the initial base outlay, and three upgrade fees to build a sensor base), and can be moved as needed.  If you want starbases that can see, build the Eyes of the Universe instead of upgrading their sensors.

Now if you'll excuse me, I need to go duck and cover.  Willy is the Influence mastermind of the forum, and I suddenly feel like lighting is about to strike me    Tinfoil hat ahoy!

I have to disagree, Willy.  I don't have my game in front of me to verify, but I'm fairly sure that base, un-upgraded economic starbases provide a 5% bonus.  I recall that looking at the planet details screen shows the 5% bonus, but the trade bonus is implied in the hint text - something like "economic starbases grant production, research, and trade bonuses.  Upgrade the base for further bonuses."  I would guess at a 5% trade bonus, but I wouldn't know how to test it.

Now, I don't like to post gibberish, but if I didn't post now I probably wouldn't.  I'll check on the production bonus later if noboby beats me to it.