How do I enable card save future as braintree credit card payment option in Magento2 on checkout page?
The behaviour of using saving credits cards (Vault) depends on your Magento version.
Magento < 2.1 has and old implementation of Braintree payment method. Is more useful and secure to use Braintree payment method in Magento 2.1.
In both cases you need to go Stores -> Configuration -> Sales -> Payment Methods -> Braintree and when select Vault enabled -> Yes.
Perhaps, this documentation will be helpful.
After refreshing Magento cache, saving credit card feature will be available on the payment step.
Note: Vault feature will be able only for registered customers.
UPD: More details in payments methods documentation.