Perhaps dealing with mines doesn't have to be a balance issue, but rather an AI issue? Perhaps if ships defaulted to 'mine clearing' as a default activity, that is that flees with scouts actively paired up to detect and destroy mines, rather than making the user do it manually, that would alleviate the mine issue greatly. Moving ships prioritize nearby mines, to 'clear the way' while ships with range to fire at enemy combatants without running into mines would do so.
I posit an algorithm to do it.
Fleets are always aware of mines, even if they are invulnerable. If there aren't any scouts, they'll move normally through the minefield, blowing up spectacularly.
If there are scouts, manuver them so that their detection radius incorporates the closest mines. If the enemy is in range, fire at the enemy. However, if the enemy is not in range, see if there are mines in the way. If there are, blow them up, them move to engage the enemy.
If there are no enemy ships in the gravity well, target mines giving them the same priority as hangers/turrents. Scouts should automatically work toghether with fleets to detect and destroy mines.
Slight balance change:
Increase the detection radius of the scout so the line between detecting mines and blowing up in them isn't such fine line, Mine detection has an unlimited Z radius.
The main difficulty is that the USER must destroy mines. No one wants to play clicky clicky all day. If fleets procedurely detected and destroyed mines on their own, then the respectively different ways of mine laying would be more balanced.
Also make it so destroyed mines don't explode. That would avoid the automatic attrition problems of players being forced to jump into minefields.
The Kosutra cannon should destroy all mines.
This is just a thing, but I don't think Vasari and Advent mines should be as effective as TEC mines, as they can be densely packed and made to be unavoidable. Right now all mines destroy all but heavy frigates and carriers in one blow. The damage done should be MUCH less, at least for free mines. Perhaps blowing up 1/4 of a frigate's hitpoints in the blast radius.
As for starbases: They don't have to be 'unavoidable', or prevent jumping to the next planet. They just have to make running through enemy territory HURT. The initial starbase is quite weak.
Starbases need to have more DESTRUCTIVE power. Sure- perhaps you can run around a starbase to a vital planet, but if you lose 2-5 frigates as you are skirting the gravwell, it had better pay off. The initial 2000 credits you spend should pay off. So if the enemy loses 2-5 LRMS, or 1-2 Enforcers while skirting the starbase, I'd say that's more than fair.
So what do you do? Increase the radius of the PJI, and the starbase so they've covering all the phaselanes. This way you are punishing the enemy for running in and out of your empire willy nilly.
Would it be too much trouble to show arcs explicitly where enemy ships can jump in and out of a gravity well?