If your paid account expires or is downgraded, or the space granted to your account by a promotion expires, all of the files in your Dropbox folder will still be available to you on your computers, phones, and via the Dropbox website. However, Dropbox will stop syncing the files in your account if your usage is now over your new quota. This means that new files added to your Dropbox won't sync to other computers or be available online until the amount of space used in your Dropbox drops below the amount of available space in your account.
For example, if you have a Pro account with 80 GB of data and you decide to downgrade to a Basic 2 GB account, your files will stop syncing because you don't have available storage space in your account. In this case, you'll probably want to upgrade your account to ensure all of your files are syncing again.