Taking into consideration the vanilla skills, at the moment, only "Fighter Doctrine level 2" would have any real effect in the spawned drones, by making them a bit tougher.
Additionally, the "Expanded Missile Racks" hullmod, unlocked by "Missile Specialization level 2" increases their ammunition count, it might not seem as much, considering the drones regenerate slowly, but the weapon cooldown is faster than their ammo regeneration time, so the hullmod allows you to fire a second wing of drones. This pairs extremely well with ships with the "Drone Swarm" shipsystem, since it replenishes their ammo count and resets their cooldown.
Additionally, in order to account for modded content; any shipsystems, hullmods or skills, that would improve missile weaponry fire rate, cooldown, ammo count or ammo regeneration, should affect the drone spawner weaponry (not the drones themselves; but would increase the number of available drones). Stuff like projectile speed, manoeuvrability, hitpoints, or weapon damage won't have any effect though. Stuff that affects a carrier's fighters won't have any effect either, since the drones are not bound to the carrier.
However anything that has a fleet wide effect should, as long as it is set to affect fighters, it should buff/debuff the drones, similarly to how the "Fleet Server" hullmod acts buffing all fighters that are close to the ship that has it.
This said, the "Hive Launcher" is an exception, since it would benefit from any missile buff, and no fighter buff, considering it is a proper missile weapon that it is set to work in conjunction with the "Drone Swarm" shipsystem.