iPad DropBox client side encryption
I use a program called 1Password for secure encryption of files (and other data) and it almost does what you ask. 

On my Mac client - it can attach arbitrary files to any entry and securely encrypt them before they get stored into Dropbox, but that isn't the primary purpose of the tool. It has an iOS client that can see files attached in this manner, but not really open them, send them to other programs and it also doesn't let you file things.

So - what you ask is clearly something that could be programmed, but isn't something where there are several well-known solutions and people will just propose several. In fact, there may not be a single solution that uses Dropbox despite how popular and prevalent it has become.

I would suggest you consider Box.net which seeks to be a more enterprise secure version of Dropbox - especially if no-one else has a better suggestion than mine for securing your files before they hit Dropbox.