Well do you still live at home with your parents? If you do you may need to have a sit down with your parents and brother. I would like to believe they would punish your brother and help you take steps to ensure the safety of both animals. Be calm while you talk to both parties and explain you find this sort of animal abuse unacceptable. If you find there is no support from your parents I would consider rehoming your betta till you are in a situation where you have more control over who has access to your animals.
Is it a roommate situation as in you are both grown and live together you may want to lock the Betta in your room when you are not home. Explain to your brother why this is unacceptable. If he doesn't get the point I would look for a new place or rehome the betta till you have a better living situation.
If he just happen to stop by and you live alone I would not allow him in your home while you are not there. I would also supervise him closely when he was there. Again speak calmly to him about this or rage. Up to you here, you are adults. Personally if it was my brother or sister I would rage. (I have raged over conditions I find appalling with my sister's treatment of animals, with good reason though. I found a dog that has starve to death in her place.)
As far as the fins keep him in clean warm water. If there is any sign of fin rot you want to dissolve 2 level teaspoons of aquarium salt per gallon and add it to your tank. The salt treatment only works in soft water as I have recently learned. If you have hard water like we do in Southern Cali adding salt is a pointless task.
Ref for salt treatment hard vs. soft water