Not really. DropBox uses OAuth for authorization to allow users to log in with their accounts. Even assuming that you figured out how to log in on your behalf with your credentials, you will transfer your credentials to anyone who wants them, allowing someone to change the password in the account, etc.
Alternatively, I suppose you could generate tokens and transfer them instead, however, for this you will have to set up some kind of web service and you will still have problems with the fact that people can do something in your account, that no doubt that you do not want them to do this. For instance. I could create an application that immediately deletes everything that anyone downloads, or I can create an application that fills your quota with files filled with zeros.
This is not what DropBox intends to use the API for (and it may actually be against ToS, you should probably read to make sure that you are following this route, despite my disappointment). You should use a more appropriate storage method.