Even though there is no official Dropbox client for Windows Phone there are some 3rd party alternatives that serves the purpose. BoxFiles is one such app that allows you to upload and manage files to Dropbox. It even has SkyDrive intergration, easy auto-upload of photos to Dropbox, etc,.
-Create a new account
-Access your files: documents (word, excel, pdf, etc.), music, videos, pictures, view and edit your text files
-Pin folders to your start screen
-Skydrive: upload files to Dropbox, open files from Skydrive, rename files or folder, create folders, delete files and folders, share links to files and folders
-Manage your files and folders: copy, delete, rename, move, create new folders
-Upload pictures from your Pictures Hub, or use Share to Boxfiles directly from the Pictures Hub, save pictures from Dropbox to your phone.
-Email links to any file or folder
-Background transfers only in the full version
-More Skydrive functionality create folders, rename file/folders, share link to files and folders, open files directly from Skydrive
-You can open .url files directly from Dropbox
-You can now syncronize the camera roll with a Dropbox folder
-Check Mode/Batch for copying/deleting moving items inside Dropbox (similar to the email client)
The app costs $1.29 with free trial available. Find it here from the marketplace.