One way to do this would be to utilize the GoogleCL tool. GoogleCL is a command-line interface (utilizing Python scripts, but the Windows version includes a wrapper so you don't have to manually install and set up Python) to various Google services, including Picasa.
(I'm assuming you're on the Windows platform for these commands, but equivalents exist on Linux)
First, download and extract the GoogleCL tools to a folder. Next, create an album on Picasa where you want the uploaded files to go. I used
Testing for this example, change it to match the one that you want to use. You can do this without using the web site by running:
google picasa create Testing
The first time that you run the command, it will launch your web browser so you can grant authorization to the application to access your Picasa web account. It will remember this token so you won't need to do that on future uploads.
The following command can be used to upload all of the JPEG files in a folder:
google picasa post Testing "C:\Users\Your User Name\Documents\My Dropbox\Photos\*.jpg"
Make sure to replace the folder path with the correct one for your system.
That command would upload all of the .JPG photos in your Dropbox
Photos folder to the
Testing album in Picasa Web. You can change the path to whatever you'd like. Note that it doesn't check for duplicates, so you'll probably want to either move or delete them from the folder after the upload command runs.
You could create a batch file along the lines of the following to automate the process (place the batch file in the same folder as GoogleCL):
google.exe picasa post Testing "C:\Users\Your User Name\Documents\My Dropbox\Photos\*.jpg"
if not exist "C:\Users\Your User Name\Documents\My Dropbox\Photos\Uploaded" mkdir "C:\Users\Your User Name\Documents\My Dropbox\Photos\Uploaded"
move "C:\Users\Your User Name\Documents\My Dropbox\Photos\*.jpg" "C:\Users\Your User Name\Documents\My Dropbox\Photos\Uploaded"
The result of that batch file would be to upload the photos in your Dropbox
Photos folder and then to move them into a subfolder called
Uploaded to prevent them from being uploaded again the next time that you run the batch file. You could set the batch file to run as a scheduled task if you wanted to automate the process.
This is a basic approach and doesn't handle all of the possible cases (for example, if the GoogleCL tool is unable to contact Picasa Web or encounters an error, it will still move the files to the
Uploaded folder), but it's a good starting point and you should be able to build on it to solve your problem. It shows that it is possible to upload a folder to Picasa Web without having to use their desktop client or the web site.
Use the command
google help picasa for information about the other options that GoogleCL supports. There are commands for listing and creating albums, so it would be possible to enhance the batch file to do something like creating a new album with the date and time as the title each time it runs if that was something you were interested in.