The answer is Yes. You can automatically backup photos and videos to Picasa Web from your computer or camera. There are two ways to backup photos from your computer to Picasa web i.e by using 'Picasa software' or 'Google+ Auto Backup'. Picasa Web is now integrated with Google+ account. When you access to Google+, you will see all your photos and videos from your Picasa web account. When you install Picasa on your computer, probably you will get the option to turn ON Google+ Auto Backup. Google+ Auto Backup is available for Windows and Mac, so you need to download it on your computer. It allows you to automatically backup photos to Google+ account from desktop computer at any time when you connect a camera, phone or storage card. You have options to select the size of the photos i.e full size, standard size and much more. By default, Google+ Auto backup is set to full size. If you change it to standard size (2080p), then you can backup an unlimited number of photos for free. If you use 'Google+ Auto Backup' to automatically upload photos to Picasa web, then you will see all possible Auto Awesome photos made by Google+ from your photos. Google+ Auto Backup works very well when you have unlimited bandwidth on your internet connection.
You can install PicBackMan's Picasa uploader for Mac from the website and start backing up photos and videos.