Downsides include increased likelihood of rain entering the passenger compartment when a door is opened, and a similarly increased likelihood of bumping one's head when entering or leaving.
A short review of the best-known production cars with gull-wing doors is below, in the form of images and captions.
Rudolf Uhlenhaut posing ca. 1953 with a 300 SL rennwagen. This racing coupe was highly successful and soon led to a production version. It introduced gull-wing doors and lent them considerable prestige as a design detail. The reason for that door design was functional: Racing 300 SLs had a tubular space-frame "chassis" that would have been substantially weakened if door openings extended down to near the lower sides of the body (see photo below showing the space-frame).
Bricklin SV-1. Its doors drop farther down the sides because it was built using a conventional chassis. I recall the Bricklin fiberglass body as appearing a bit crude, having thick-seeming panels. If others got the same impression, that might have affected sales. 1974-76 Bricklin production was 2,854.
DeLorean DMC-12. Around 9,000 were built during 1981-83.
Giorgetto Giugiaro of ItalDesign.
Clearly, cars with gull-wing doors were never very successful in the market. The 300 SL was very expensive, which held down sales. But it lent significant prestige to Mercedes-Benz during its recovery after World War 2. Bricklin and DeLorean, like most automotive start-ups, were under-funded, which was probably the most significant factor in their failures. The gull-wing doors might actually have been a plus, because the cars were sports cars and buyers of such cars tend to more tolerant of inconveniences than those wanting more practical conveyances. The DeLorean had better build-quality than the Bricklin plus a lot more media buzz thanks to the charisma and General Motors background of DeLorean himself. Malcolm Bricklin's previous automotive experience had to do with being the first to import Subaru cars to America.