Drupal module updates failing through administrative interface after update to 7.50
This is most likely a problem with your file permissions. A quick test on simplytest.me showed that module installation is not broken in 7.50.

I would try to check the file_temporary_path variable under:

"Administrator -> Configuration -> Media -> File System"

And see that the web server can write to the configured directory.

If my assumption is correct, then also all other file uploads will fail, so it might be a good idea to test uploading an image to an article, too.