The Blog This button (also under the menu "Create" > Publish To Blogger) both used Picasa to call transport service to Blogger. The underlying transport to the services was changed from http to https and the URL in Picasa was hardcoded as http in the custom button files.
The BlogThis! function in Picasa no longer works. Here is a work-around.
Use the Blogger web interface
Create blogposts in the webinterface of blogger.com directly following these instructions:
Sign in to Blogger
On the left, click the Down arrow Down arrow
Click New blog
Enter a name for your blog
Click Next
Choose a blog address or URL
Click Save
More info: https://support.google.com/blogger/answer/1623800?hl=en&ref_topic=3339243#write
Tips on using Picasa with the Blogger web interface:
The steps below show how to use Picasa to select and format photos for upload to Blogger
Start Picasa
Using Picasa, export the photos to be uploaded to the blog to a folder on the PC:
If you need (for example) 10 pictures in your new blog post, select the (already edited) pictures.
Select the Export button at the bottom of the Picasa screen.
In the Export screen set Image Size to 1600 using the slider.
Set Image Quality to Normal .
Remember the Export Location for future reference, then click the Export button.
Note: Unless you change it, the export location will be: Pictures\Picasa\Exports\XXX
Where XXX is the name of the Folder or Album the photos were exported from.
If you never change the export location it will be easier to remember it in the future.
Now start your Blogger Dashboard and start a new post.
Then click Insert Image > Upload > Select files > search folder XXX (where XXX is the Export location you remembered above).
Select the first thumbnail and then press Ctrl+A to select them all. (or select the ones you want in the current post)
Click Open and the selected photos will start to upload very fast because of the reduced file size.
After uploading is complete you can place them in your empty blog all together (one below the other with some spacing for text).
Then you can adjust the size and position if necessary and you can add text as necessary.
When the post is formatted to your satisfaction, it is ready for publishing.
If you have published all the photos you need in the Exported folder, you can go back to Picasa and delete the exported XXX folder because it contains duplicate photos.