Right now my main method of dealing with minefields I can't just go around is to build a big fleet of Seeker Vessels and just suicide them through the mines to clear a path. Unfortunately, this is expensive as balls, and very time consuming. I'd much prefer another way, if anyone knows one.
Personally, I use my titan as a giant mine field bulldozer! It can usually take the abuse without being destroyed. If I don't have a titan yet I try to use capital ships or other sturdy vessels. The other thing to keep in mind, is that the other players (Human or AI) have the same problem with minefields, so don't forget to use that to your advantage.
Your other option is to sit around and wait for an update that fixes the problem.