Well as the title says, Starbases should be moveable similar to Shipyards.
With 1 tile per turn or something similar and the same anchoring/unanchoring mechanic like Shipyards with temporarily suspending whatever the starbase is doing.
Let me explain why this should be a thing:
1) Placement of Starbases is sometimes just a chore because of how one or more resources/relics are just 1-2 tiles out of range. While there are technology upgrades available later (namingly "Support Field Stabilization" and "Exploration Extension") which expand the range of the Starbase up to 4 tiles) for a total range of 9 tiles when maxed out, it doesn't come in handy at all because at this point you already placed 2 or more Starbases to make up for the limited range you had the first time around (especially when you are new to the game and didn't know about the later technology) so you leave it be as it is since later on the module doesn't make much of a sense anymore.
When moving the starbase would be a thing then one could place the Starbase near the more important resource/relic and later upgrade the starbase with the extended range module and move it to a little bit better spot to cover the resources you couldn't reach earlier, which eventually gives the modules much more sense at that point, so you don't have to think too much about the placement early on or wait until the tech becomes available. It would help to cut back on micromanagement costs since 1 Starbase covering everything is better than needing 2 or 3 starbases for doing the same. One would willingly invest more into existing starbases rather than just plopping new ones everywhere.
2) When the influence of an AI/other player reaches the Starbase and gets overrun by it then they consider it their territory and are more eager to attack the Starbase up to the point where one would have to excessively defend it for just a few unimportant resources which might not be worth the effort leading to an all out war. At some point you'll waste resources to defend it or just give up on it and look for a better spot.
When moving the starbase would be a thing then you could move the starbase elsewhere, rather than completely destroying it and thereby not waste on the time and resources you invested to build and upgrade that starbase. You could keep it and move it to a strategically better position, altough very slowly and thereby satisfy the otherwise neverending complaints of the AI. Even Deep Space 9 has done that several times throughout the series.
3) One might argue that moving Starbases is unfair to the AI/other players, but it is NOT:
The thing is, when moving would be a thing then hypothetically you might build a heavily defended Starbase and then move the Starbase into enemy territory and start spreading your influence there.
Seems unfair at first, BUT currently one could do basically the same by sending a truckload of Constructors into the enemy territory and build a new heavily defended base as well so currently nothing prevents you from doing so anyways. Balancingwise this nothing different than moving an already existing base there would be.
And even if it would be a balancing issue for some, then I suggest that building a Starbase in an other players/AI's territory should be impossible anyways, and so should be moving a Starbase, if that ever becomes a thing. One should only be able to retreat a Starbase from enemy territory to your own territory if their zone of influence overtook your Starbase.
TLDR; Make Starbases moveable so the extended range modules (namingly "Support Field Stabilization" and "Exploration Extension") make more sense for existing starbases to cover nearby resources that where out of range earlier on, as well as to strategically retreat a starbase from enemy territory into your own territory so one can avoid an unnecessary war without having to destroy all the work one has put into it.