It depends upon what caused the dropsy. As I said, dropsy is a symptom, not a disease itself. It is a sign that something else is going drastically wrong with your fish. It's like a runny nose and a cold. The runny nose is not a disease, it is a symptom of your cold. You just normally don't die when you have a runny nose.
Dropsy is caused by several things. The most common being overfeeding which leads to bloat, and then progresses into organ failure. This takes several paths. One being the stomach and intestinal tract becoming kinked and pinched off because of too much food, which then causes the digestive tract to become necrotic (aka dead tissue). They can also simply become bloated, and will display SBD symptoms, primarily because the swim bladder is being pushed over to one side of the body because of the bloated stomach, and can also sometimes become inflamed because of the rubbing of the stomach against the swim bladder (organs are sensitive, they do not appreciate being rubbed against each other). Occasionally this even progresses so far as to cause a tear in the stomach or intestinal tract. Once that happens, unfortunately, your fish is screwed, and will be dead in a day or less.
This cause can be easily prevented: don't overfeed your fish! A proper diet will prevent this from happening.
Kidney failure as well as liver failure is know to cause dropsy, and the kidney failure is generally caused by a bacterial infection. A bacterial infection often caused because of the diet. Certain live foods could have been carrying the bacteria (I hear tubifex worms are notorious for this, I don't feed them to my fish myself, so can't say from personal experience), or sometimes (this is notorious for happening in pet stores) the pellets become wet when being stored, and rot, growing a whole host of bacteria and fungus. Instead of being thrown out, they continue to feed this to the betta. Basically, it's food poisoning, fishy style. They kidneys try to filter this out, become overwhelmed, shut down, and die (remember that necrotic thing I mentioned earlier?). And dead kidneys = dead fish. So does dead liver.
This can sometimes be cured: If you catch this early enough, and treat it aggressively, you can cure the bacterial infection. However, the key is catching it EARLY, because it doesn't matter if you've killed every last one of the bacteria if the fish has already lost its kidneys or liver. You can't live without them, neither can your fish. Generally by the time you spot the dropsy, sorry, you're out of luck, it's too late, the fish is already in organ failure.
Parasites can function similarly, in time overwhelming the betta's immune system, weakening them to a point where various organs start failing, and they eventually succumb to the parasites, which have generally by this time colonized inside the fish's organs. They often attach themselves in the gill arches or to the heart as these are sources of high blood flow, which mean easy nutrients for the parasite.
This can sometimes be cured: Catch the parasites early and treat aggressively and you may get lucky and save your fish before they are irreparably damaged. Careful though, parasites are stubborn things, and even if the betta survives this brush with death, they will generally always be much weaker and much more susceptible to diseases of any sort in the future.
Tuberculosis is also a known killer of fish and a cause of dropsy (though you'll also generally notice these fish will develop popeye too). Unfortunately, this is 100% fatal. Why? Because we can't cure tuberculosis. It will kill the fish every time, no exceptions.