Let's consider for a moment just how large a volume of space each tile represents. According to this post by Draginol (Frogboy), the minimum dimension of a tile is ~0.02AU or ~10 light-seconds. If we're very, very generous, GCIII ship combat ranges might be roughly 0.4 light-seconds (beams, which include lasers, take about 0.2 seconds to propagate from point of origin to target when fired at maximum unboosted range, and the maximum possible missile range is roughly double the unboosted beam range; you could also argue for engagement ranges more like tens of meters to a couple kilometers based upon the sizes of the ships and the apparent distances between them as seen in the battle viewer, and put the laser component in the pile of science fiction 'lasers' which aren't true lasers). If you're very, very lucky and happen to be mining the minimally-sized tile, you might be looking at ~600 mines per sheet across the tile, with several sheets of mines required to effectively block travel through one face of the tile for any significant length of time (moving parallel, or nearly parallel, to the minefield is not going to be hampered). If you want to block all travel through the tile, you're probably looking at something more like 16000 mines just for the smallest possible tiles in GCIII. If you assume that the average tile size around a star with a system similar to the Terran home system is ~1 AU (and in my opinion the average tile dimension is likely far greater), you're looking at needing ~2500 times as many mines per sheet (~1.5 million mines per sheet; roughly 100 million mines per tile with 25 sheets per field using two or three mutually perpendicular fields to block passage through the tile, or ~2 billion mines per tile if you want to fill the tile). Once you get out into interstellar space, the tile dimensions probably jump up to something on the order of a lightyear rather than an AU, at which point you're looking at something on the order of 1e15 (one quadrillion) mines per sheet across the tile. Mining a full tile, even with generous assumptions about mine spacing, requires an enormous investment of resources and a lot of time. This is quite simply not practical, especially if you want to apply it to every tile in a starbase's area of effect (without AoE upgrades, a single starbase covers 90 tiles, not including the tile in which the starbase is located; the first AoE upgrade increases this to 168 tiles, and adding the second increases this to 270 tiles).
Then there's the issue of whether or not the mines are sufficiently difficult to clear to actually provide a practical impediment to a ship's progress. Unless you have near-perfect containment, anything carrying a significant amount of antimatter is going to be emitting fair amounts of high-energy radiation, which should be relatively easily detected and strongly indicative of the presence of something unnatural. Anything carrying fissile material is likely to be radiating due to the ongoing radioactive decay in the material. Anything with an active station-keeping system (which would likely be necessary for mines in free space rather than in orbit around a body) is going to be quite visible and easily distinguished from natural space debris. Anything with an active power supply is likely to be fairly detectable and readily distinguished from natural space debris. If the mines are readily detectable, that removes the single greatest obstacle to clearing them; now they have to fall back on being difficult to target or present in such great numbers that clearing them is very time-consuming.
There's also the issue of whether or not ships travel with inactive defenses, whether or not a mine is capable of penetrating those defenses if active to a degree sufficient to be worth the cost of the minefield, whether or not the time between the mine detecting a target and detonating is long enough for the target to detect and respond to the mine's activation (you're proposing a mine which orients itself towards the target; it's going to be quite visible for at least a short time before it detonates), how many mines a ship must trigger before the damage sustained becomes problematic or causes the loss of the ship, and so on.
There are scenarios where space mines can make sense, but Galactic Civilizations III is not in general a setting in which space mines make sense; at best, you might do something like set put up an orbital minefield around a planet to deny certain orbitals or impede passage through certain orbitals, but this is going to have detrimental effects on civilian traffic to and from the planet and is unlikely to be something you can set up rapidly enough that you'd only have the minefield while the planet is under threat of invasion, and that minefield would be in the same tile as the planet, not something impeding passage through an adjacent tile.