On rare occasions, people have contracted fish TB from fish. It manifests itself mostly as infected sores on the hands. I've never used gloves in my tanks which I'm pretty sure all carry fish TB. I've never had symptoms but others have. Many times, I've had cuts while in the tanks. It's highly unlikely you were affected especially since you have no cuts but if you notice anything strange on your hands, see a doctor and mention the fish TB. If you're worried, you should wear gloves when working on the fish tank.
Here's a page with some basics about fish TB including humans getting it:
http://www.4qd.org/Aqua/disease/tb.htmlDropsy is rarely cured but you can try aquarium salt and antibiotics. I hope your betta beats the odds and improves.