Four separate ways to deal with a drone ship, among others:
1) Most common: Shoot the drones.
Webs, blasters, warrior drones and a strong tank help here. Can be a struggle if the target has a huge drone bay. If your guns have poor tracking but good range kite at high speed to get the drones to line up behind you with low transverse. This also keeps you out of range any high-slot weapons.
2) Most amusing: Sensor damp the ship before it gets its drones out.
Vexors and VNI’s have only average targeting range: two remote sensor damps in a cheap Maulus frigate will bring targeting range down from 65km to <20 km. So fit sensor damps, a warp disrupt and sit at 20km. Drones that were out prior to the damp will target you if set to hostile - but only when they finish with any NPC they may be attacking. New drones released after the damp will not target you regardless of their setting.
3) Most sensible: Speed tank the drones.
Drone damage decreases with increasing speed: my tests suggest drones do approximately half damage at half their maximum speed, only ~30% damage at ~75% drone maximum speed, and they struggle to keep up and hit at >80% of their drone max speed. Large and medium drones should not be able to hit an orbiting MWD frigate. Hob II’s struggle at >3000m/s, and Warrior IIs struggle above 4000m/s. Kite outside scram and neut range (>12km) in a fast ship and blap away.
4) Most prepared: Shoot his ship and tank his drones
If the target only ever carries 1 race of drones (check his kill board) fit your ship to tank their damage type and attack his ship while ignoring the drones. This assumes his high slots don’t carry much risk.
You can also have a go at using ECM.
In reality a combination of the above works the best: kite and speed tank while shooting drones you are tanked against.