Keeping threat low is very useful. When your threat is low, you don't need to worry about threat balls (the name for such piles of threat ships).
One method is to avoid gaining threat. To do this, you need to know how threat is gained, and what threat even is. I'm guessing you probably know some of this, but I'll cover as much as I can to try and be useful to anyone with questions on this. I also don't know how useful this will be to your present situation.
Threat are ships that have been freed (aka, released) to attack you whenever they like. They are generally the non-greyed out ships on the map (some non-greyed ships are something other than threat, such as Special Forces or part of an Exo Wave). In general, once freed, a ship is freed permanently. I can't think of an exception to this, but I'm sure there is at least one. So, how and why are ships released:
1) You attack an AI planet and upset them. If you attack with an overwhelming force, all the ships will be released in a system. If you destroy the AI Command Station, often all the ships will be released. If your attack force is smaller, only ships you get in range of (either you shoot them or they can shoot you) get released. In general releasing happens by Guard Post. So you'll release all the ships at a Guard Post, or none of them. This means that if a Guard Post has even one really long range ship (Zenith Bombard for example) they are much easier to trigger. The Guard Post shooting matters as well, so Missile Guard Posts are very easy to trigger.
So once you've woken up the AI ships, they'll generally either attack you or flee. They normally only flee if you outgun them. This can cause a problem if you attack with a huge force, because you wake everything up and then cause a lot of them to flee, leaving you with threat hiding out on other AI systems. Attacking with a smaller force to wake up ships piecemeal and grind them down causes less threat to escape. You can also use a small attack force to kill the Command Station and you'll often get all the ships in the system converge to attack you. Lead them back to your worm hole and bring the rest of your fleet in to finish them off before they can flee.
2) Retreating waves result in threat. Waves only retreat when they feel overwhelmed, and they are supposed to stick around at least 30 seconds. But often players have things set up so the waves are vastly outgunned, so they will retreat at 30 seconds. To counter this, get a Black Hole Machine setup on your whipping boy or build several Force Fields over top of the worm hole waves are coming from. This pins the wave in long enough to get destroyed. Underneath those Force Fields is also a handy spot to drop some Lightning Turrets, Flak Turrets, a Gravity Turret, a Tachyon Turret and possibly some Tractor Turrets.
3) CPA always produce threat, a lot of threat, so you need to learn how to clear it. Nothing you can do about it. Around every two hours or so you have to handle it. If your defenses are strong, a CPA is just threat creation. If they are weak, it is attacks from multiple locations. CPAs and existing threat is a really bad combination, so always try and clear out threat before a CPA hits.
There are other reasons to clear threat quickly. Threat ships are own their own for 30 minutes. Then they group up with other ships that have been threat for a long time and form a threat fleet that will cause you tons of grief. So, clearing threat quickly is much easier than waiting, especially for large amounts of threat such as produced by a CPA. If you have scouts on all your adjacent planets, in the Galaxy Map you can change the filter to view threat and see exactly where all the threat ships are. This lets you take in your fleet ships to clear them out. Since threat ships generally sit on your worm holes waiting to attack, you can often decimate them fairly easily. Be careful about sending in too many of your own ships or you can trigger a lot of ships to be released in the system.
Another fun trick is to use Riot Starships with Tractor Beams. Pop through the worm hole and come right back, dragging all those threat ships camping the worm hole to their death at the hands of your waiting fleet. Other mobile tractor beam ships can do this too, but Riots are always available. Give this a try and see if you can prune the threat back. If your Riots get too beat up, try sending in a cap of Mark I Fighters too. They'll absorb a lot of hits while your Riots bounce in and out without costing you much to replace.
