There are different ways to backup photos to Dropbox. The Dropbox mobile app makes it easy to backup your iPhone photos. Once photos are uploaded, they will sync with Dropbox on your computer and all your other devices.
Download the Dropbox app first and create an account if you don't already have one.
Launch the DropBox app
In the Dropbox app, go to Settings > Camera Upload > On.
Select whether you want Dropbox to only upload pictures while on Wi-Fi (recommended) or while or Wi-Fi+cellular (not recommended if you have a capped data plan).
Tap "Enable." Dropbox will ask permission to access your photos. Say OK. The app will then start uploading all your photos.
It also allows you to manually upload one by one instead of uploading all at once.
You can install PicBackMan's Dropbox uploader for Mac from the website and start backing up photos and videos.