Breeding bettas lowers their lifespan and puts them at risk for illness and even death after spawning, so treating them after spawning is a high priority, even if you don't see anything outwardly wrong with them .
The most common injury is torn fins. This can happen from nipping, but also if a betta swims quickly across the tank their fins can get caught on things and tear. This mostly happens to females. Other injuries can include bruises, scrapes, missing scales, and more. In the most extreme case, I've had a female betta injure a male so badly during spawning that he passed away shortly after attempting to breed. This is a reinforcement on why choosing bettas with the proper temperament is so important.
The first thing to do is move the female to a hospital tank by herself. If you do not have a hospital tank, then a 1 gallon heated jar will do. Even if there is no visible injury or ripped fins, it is important to observe her for 72 hours.
Raise temperatures to 80-82 degrees. I find that this helps boost their metabolism and promote healing.
Keep the water pristine. Do daily 25% water changes, and 100% every 3 days if you need to keep her longer.
Add Tannis through Indian Almond Leaves (Catappa Leaves). If you have none, Oak Leaves with no pesticides will also do. You can also add liquid tannin.
If there are ripped fins or other signs of injury, add 1 TSP Aquarium Salt per gallon. And after 3 days do a water change.
Feed her 2-3x a day with high protein foods. Live foods are my favorite, followed by frozen Blood Worms.
Observe for any signs of infection or behavior changes. If spotted, use appropriate medication for her infection.
Once she seems healed, put her back in her tank!
For more advanced injuries, check out my sick betta page. (WIP)
If a male is injured during spawning, it means that something went wrong. Males should not get more then a torn fin or so. Either the female was too aggressive for him, or he was too timid to breed.
A male should not be taken from his fry, and treatment cannot be added to the tank, so the best course of action is to provide high protein food and make sure the water quality is pristine. Personally I do not feed my males live foods while they have fry, opting instead for frozen or quality dry foods.
He should only be removed from the spawn tank early if he appears to be ill. There are ways to hatch a spawn without the dad, but typically the turnout for fry is a lot lower then if you keep him in.