Updating Safari Browser (macOS)
Open the App Store
Click the Apple menu icon in the top left-hand corner of your screen.
The system menu will appear, click the "App Store..." item.
This starts the App Store application.
Go to the Updates tab
Along the top of the App Store window which has just opened will be several tabs (or buttons). Click the one on the far-right named Updates
Find and activate the Safari update
(NOTE: The Update screen may look different to the one in the screenshots. Version numbers shown here are examples and may change depending on what is considered the latest version)
On this screen, the App Store will show you all of the updates which are available to you.
It will bundle all of the System Applications (eg iTunes, Safari, Security Updates etc) which have updates available into one main "Software Update" item - shown in the following image.
If you click the UPDATE button, it will install all of those updates in one go. Notice that in some cases this may also require you to restart your computer.
If you click the UPDATE ALL button, it will install all of the System Application updates, as well as any Third Party updates - you may be required to restart with this option as well.
It is generally a good idea to install all of the available updates, but be aware: this may change how some of your software functions in the future. You should always have a backup of your system before you install System Updates.
Upgrading just Safari
If you want to upgrade just Safari and install none of the other available updates, click the blue "More" link; this will expand the System Applications, allowing you to pick and choose which updates to install.
An expanded list of software which has available updates:
Choose which update/s to install by clicking either the Update or Update all button.
App Store will now update Safari on macOS
The latest version of Safari available for your system will download and install itself. This may take a few minutes depending on the speed of your internet connection.
You can see the blue progress bar showing how much of the installation has occured.
Safari is now up to date
Safari is now shown in the Updates installed in the Last 30 Days section of the App Store Updates.