Place the final date of each iteration as a Milestone in Basecamp. That way you can keep track of how many days you have left. Burndown uses that information to know when to stop graphing.
Use Basecamp's to-do lists as a sprint backlog. Separate each User Story into tasks, and make each task a to-do item. You can estimate the amount of time or effort that each task has by adding a number at the end of the description of the task. Burndown will take that information later on to figure out how much effort you need to finish the iteration. Remember to assign all the user stories to the milestone that represents the iteration to which they belong.
Create a new burndown chart in Burndown by clicking the new burndown button. Pick the project and milestone that represents the iteration for this burndown.