Mines are, by default, in phase space...this makes them invulnerable to damage...

All scout ships have an ability to "decloak" mines, allowing them to suffer damage...

ANY ship can damage a mine, but only if the mine has been decloaked by a scout ship....

The easiest way to deal with mines is to have a few scouts scattered around the edge of the minefield so that all the mines are decloaked...then, SC are usually the most efficient way to destroy the mines once they are made visible...

Strikecraft are fast and don't suffer from exploding mines, so they should be able to destroy the mines quickly and without loss....however, you may want to spread around the squadrons of strikecraft so they are shooting at different mines....SC can suffer from "overkill" where all the ships shoot at the same target and at the same time, doing way more damage than needed and wasting their shots...

Homing mines are the only real problem.  If you don't have anyone shooting at you while you're doing it, just set some scouts loose in the gravity well and they'll clean things up.  Homing mines are very annoying and require you to carefully inch forward to reveal them with scouts, then blast them with fighters.  The diplomacy cruiser is definitely a preferable way of clearing them if you have Diplomacy (though that's a bit of an exploit in and of itself...)