Thanks very much for the feedback; it's really appreciated.
I'm curious what version of the game you're running. Is it 1.013, or is one of the prereleases? If a prerelease, which one?
I'm a little alarmed that you're having such troubles, particularly on 6. Difficulty 7 is where the AI gets most interesting, but I'm worried that things are getting hard enough that people won't be able to play a game of reasonable difficulty on that setting. The recent prereleases have made the game even harder in many respects, with waves that come in with hundreds or thousands of ships (particularly in single player, I agree with liq3 that it's hitting "silly" numbers and that it's made worse by fighters being the most durable and irritating units around), while attempting to balance that with more income. This can be overwhelming and demands a lot of defense; personally, I think it's rather difficult, but I've been playing mostly on 8 lately; perhaps I should be bumping it down a bit.
It sounds like the problem for you isn't so much the vast quantities of ships from raids, but rather cross-planet attacks. They can be dangerous and are triggered by too many ships on a planet, which will typically be your neighboring worlds. It's really important to keep an eye on neighboring planets to ensure that they never get enough ships for this to happen, especially on III or IV worlds, where a cross planet attack is especially devastating. If you're doing this already, though, then I'm even more concerned.
I'm sure Chris will have a good response in the morning when he sees this and could look into tweaking things if necessary, but in the mean time, I'll at least try to offer a few ideas for you. Harvester exo-shields are pretty knowledge-cheap and are a fantastic investment on border worlds, making your harvesters basically invincible against anything but fleets of bombers or any ships that can fire through force fields. Also, force fields in general are definitely your friend, and throwing them over the far side of a wormhole protects you from enemy ships coming in through that wormhole (again, unless they're immune to force fields, or unless you have a large fleet of bombers or massive fleet of other things). It's really helpful keeping resource stores handy so that you can pump out ships if need be, either to replace recently destroyed fleets or to 'move' a distant fleet (if you're hitting ship caps) to the battlefront by deleting that fleet and building a new one locally as quickly as you can. Also, in the 1.014/1.201 prereleases, starships are significantly more powerful and, in my opinion, a very good investment. Leech starships in particular are great. Starships are a lot of bang for your knowledge buck, though they are pricey on metal and crystal and take a while to build (several engineers are basically necessary).
Regarding turrets, I don't think they're a very good investment unless protecting something important (like a wormhole or command station) and backed by a force field; I feel that the knowledge is better spent on improved ships. The number of incoming ships is simply so high lately that stationary defenses hold little value to me, and tractor beam turrets are too expensive for the number of ships they capture compared to the total number in a raid and take too long to build compared to the length of raid warnings for them to adequately protect against raids. Others may have differing opinions, however. If you're going with turrets, lightning turrets paired with tractor beams by wormholes is probably a great way to go.
Finally, if you're playing the prereleases, keep an eye on the threat info at the top of the screen. It'll help keep you in the know regarding what to expect on your border worlds.
Again, thanks for the feedback. We're happy to discuss this further, and I'm sure that Chris and perhaps others will have some thoughts on this in the coming hours.