So a big problem is people carpeting gravity wells with mines, right?
How about having detonating mines damage other nearby mines? If you get the range and damage dealt tuned properly, you can clear tight, large clumps of mines easily, because they'll set each other off. But a more reasonably sparse field of mines would have to be swept one mine at a time. Small clumps (3-ish?) would be okay because there aren't enough of them to set each other off. (say, 10 damage per detonation but 30-40 hp per mine?)
A guy charging into a dense field still gets ripped apart, because he'll be in the middle of the carnage, but if you run into a system with a zillion mines globbed together, you can just fire into it and set off a chain reaction to clear them.
It even makes some realistic sense.
Otherwise, the "Free" mines should probably not be free -- make the mine layers cheaper and make the individual mines cost some small amount of resource so people actually only put them where there's some tactical sense to it.
Another idea is to make dense minefields negatively affect friendly forces -- probably just slow them down as they navigate the mines once they get too dense. That will force players to limit their mining, or leave travel lanes open. This is more attractive if you can actually hide your mines from the opposing forces. They can't see the open travel lanes, so they'll still either hit the mines or take forever scouting out the lane while you respond to them.
Both approaches would take some trial and error to get tuned and balanced properly, but that's what a beta is for, right?
I disagree with some other posters that say mines should take tactical slots -- the biggest reason for this is that if you *require* tactical slots, you can no longer lay mines in uncolonizable systems. And not being able to mine the empty system that leads to my more valuable systems just feels horribly wrong. Make it harder for me to do, sure, but I really want to be able to put defenses in systems in which I have no planet. I should also be able to lay as many mines as I can afford. But then, I've always felt I should be able to build as many orbital structures as I can afford, too... capping the possible benefit from them made more sense to me than "running out of space".