Anti-drone tactics?
My preferred strategy at this point is to use a combination of burst laser and ion to keep enemies' weapons offline, however I seem to run into a lot of problems especially with higher level ships if they have drones. They tend to beat down my shields and if the opposing ship gets a good volley off it can really wreak havoc especially if I am dealing with a boarding party as well.
I have tried to keep drone control offline on the enemy ship but it doesn't work as well as weapons suppression (as above). I think this is mostly because drones come back online and begin attacking immediately after the system is repaired, as opposed to weapons needing to charge.
Is there any way to destroy or disable drones other than damaging/disabling the enemy drone control module?
Solution 1:
As always, the best defense is a good offense. If you can take his drone control offline, you just need to work harder to keep it offline. That might mean targeting the drone system to disrupt repairs even when there are other undamaged systems to target. You are correct that drones can come back online quickly and this is an issue. I'm not sure the enemy actually has to spend drone parts to do this, though you would have to.
Defensively, dodge and shielding of course matter. You can't have too much of either (well, dodge from engines experiences diminishing returns at higher levels of system improvement, so don't overdo it there). Getting to 2 points in shields fairly early can be a very good investment, and I like to have 3 by sector 5 or so.
You can destroy drones around your ship with ship weapons, however this is a coincidental thing and the drones can be brought back online with the expenditure of a drone part. If the orbiting drone happens to collide with a projectile, boom. It's very unusual; I have only seen it once myself.