If you don't find the credit card details on a reservation page imported from Booking, these can be the possible reasons: 

• Reservations made on Booking before the connection: it's a security policy from the channel, so only the main data of the reservation are sent to fill out the calendar of the system. If you accept credit card as a payment method, you'll get the data normally on the confirmed reservations after the integration with Stays. 
• Your Booking listing accepts reservations without credit card: this happens because the listing is set to accept reservations without asking for credit cards details, specially for domestics guests. Learn how to change it. 

For American Express cardholders, the security code is not requested by default of the channel. This setting can't be changed on the channel or on Stays.

The contexts above usually happen when the user already have a listing on Booking before connecting to Stays. If you're creating Booking listings via Stays, you must set the payment methods from the channel's settings, on [Channel Managers Dashboard > Settings > Payment].