It's theoretically possble to have 24 starbases bosting a single planet, but the planet has to be in a very specific location to do it. For most planets, 16 is all you're going to get.
The bonus to trade simply isn't worth the investment in that many constructors - it will take several game-years to recover the money dumped into them, and the whole system can break down if someone is forced to declare war.
The planet prodution boost is another matter. Adding 24% to industry and research per base is pretty good. Adding 384% to a group of planets is enough industrial power to take over a galaxy.
Military bases boost the power and speed of ships in their area. Piling a bunch of minimally armed and defended ships under either 16 or 24 of these is what we call a military starbase array. It's used for building a military score, either to intimidate the AIs or to inflate scores being posted on the metaverse. Military bases can be used in general combat, but the hefty constructor cost and stationary nature of the base make for very few situations where they are cost effective.
Economic bases boost trade income, but only for routes that pass through the area, and only while the mini freighter is in that area. Rarely is this bonus worth the cost. Their other use is to boost the production of planets in their area, and that can add up very quickly. Just make sure you have enough of an economy to pay for it all.
Influence starbases do NOT increase the influence of planets nearby. They generate their own influence, and the modules added to them modify that generated influence. The official ring marking their area is misleading as well; unlike the other bases, the effects of an influence base taper off over distance. The most effective place to put an influence base is in contact with the target planet - either someone else's planet you want to flip, or your planet you want to prevent from flipping. These can also be used to open a "bubble" in Yor influence, to sidestep the effects of their Super Isolationist speed limit. As long as your ships start their turn in your territory, the speed limit doesn't affect them.