How do I stop the collection emails from going out?

Charges that fail to go through are put in a dunning queue that is shown on the top of the Payments screen.

You can remove an attendee from receiving the dunning emails: 

1. Event → Registrations → Payments → Declined Payments
2. Click 'clear' next to the attendees name

If the attendee paid by cash or check at the event, or you are applying a credit or discount to their registration, you can enter it as an offline payment. That will also clear the debt and stop the dunning emails from going out.