Why doesn't the A320 have a dorsal fin like the 737?
Boeing learned the value of a dorsal strake the hard way: On their civilian version of the B-17, the Boeing 307 Stratoliner, the first Boeing aircraft with a pressurized cabin, the lack of a strake ahead of the fin caused the rudder to lock in the hard-over position when the pilot demonstrated the capability of the aircraft to fly with two dead engines on one side. The aircraft entered a spin and crashed, and that was on a demonstration flight for prospective customers.



Early Boeing 307 with the two right engines off and props feathered (picture source). Note the deflected rudder and the rectangular windows in a pressurized fuselage. It worked because the pressure difference was small.



Late Boeing 307 in flight (picture source). The vertical was enlarged and had the strake added; the same vertical was also used on later versions of the B-17.

With today's hydraulically operated controls the risk of an uncontrollable hard-over is reduced, but exactly this happened to several early Boeing 737s. From the 737-300 on the strake was added and the control system (PCU = power control unit) was exchanged for a  modified version.

Airbus never had those issues and did not see the need to add a strake since the existing vertical could do its job in all conditions. A strake is worthless at small sideslip angles and will only help when the vertical is close to stalling due to extreme sideslip. Adding a strake will increase the wetted surface area and add friction drag, so Airbus did not include one.