You won't be able to see when sensitive files are downloaded. This can cause compliance concerns. Also, if a disgruntled employee leaves, you need to shut down their dropbox right away -- but you might not know what they downloaded or accessed before/after their exit. So for instance, if they have dropbox on their phone, they might get the files before you lock down the account. And you wouldn't know what files they downloaded.
For your longer term question milton, here's a quality guide to a file sharing server (FTP) alternative that I think is better served for file sharing for organizations ...-- pdf download
The alternative recommended there is SmartFile -- I use them. They have a cloud service and a server you can put on site (to replace your file sharing server).
What will be nice for IT is you won't have to install it on all the devices and they help monitor all your files and user activity, such as failed login attempts or deleted/downloaded files. You can even integrate dropbox with it if you really wanted to in order to add the monitoring.
The API works with tons of languages and is one of the best in the file sharing industry.
So if you want to go to dropbox, just keep in mind the concerns I listed at the top. I'd recommend looking at other options though given your situation.