It seems like Dropbox doesn't give any options to remove or change payment info. I'm reaching out to the community because I need help with this issue! It is also quite annoying that email support is a paid feature.
UPDATE: I have reached out to support via email, and they replied (I thought it was only for paid subscribers...). Basically, they told me that payment methods get automatically removed once you downgrade to the basic plan. I was already on the basic plan, but after they replied, the payment info vanished from my account. I don't know what actually happened, but the issue was solved.