The CVV field only renders when CVV verification is turned on for your account. Go to Settings, then Processing, then CVV in the Braintree control panel, enable the CVV rules, and generate a fresh client token. Drop-in reads the token's configuration when it builds the form, so the old token without the CVV rule keeps hiding the field until you replace it.

Context: Stack Overflow #25148517 (accepted answer, 29 votes): The reporter integrated Braintree's Drop-In UI for recurring payments and the form only rendered card number, expiration, and the PayPal button. No CVV field appeared. They assumed CVV was optional but could not find any setting or docs explaining how to make the field show up.