The way the curse itself works on the cursed children actually heightens their immune systems along with making them far stronger than an average human.
It’s extremely hard to get them sick with a common cold or the flu or even regular viruses. Not impossible, but as close to it as they can get. Their bodies fight off the infections long before they notice.
Getting drunk is far easier, and their own damn fault if they do it (smol Dean could use some lessons in portion control when it comes to drinking).
Of course, living most of their lives without getting common illnesses also leaves them incredibly unprepared if someone does get sick, the way Sam had no idea how to tend his older brother when Dean made the mistake of drinking until he got the worst hangover.
They’re much better at this now with some schooling from Jacob.