Upgrading Operating System.

As far as I know, you can update any Mac that support Sierra to at least High Sierra.

Whether you can update your Mac beyond that will depend on its model and hardware release year. (You can find that information using the About This Mac item in the Apple () menu.). If you post that information (no serial #s, please!), we can look up how far you can upgrade.

If your copy of the High Sierra installer is too old, its security certificate may have expired, in which case you will be told that it is damaged when you try to run it. If you run into that problem, use the High Sierra App Store link

How to download and install macOS - Apple Support

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I would suggest making a bootable clone backup of your existing Sierra system before attempting any upgrade … both to protect your data, and because I'm not sure how reliable Internet Recovery / installation of old versions of macOS (like Sierra and High Sierra) is these days.

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High Sierra probably has a newer version of Safari than Sierra does, but that version of Safari is also old, and your husband's banking site might not want to work with it, either. Firefox 115 ESR runs on Sierra and High Sierra, and may let you access some sites that older versions of Safari would not.

Mozilla Support – Firefox users on macOS 10.12, 10.13 and 10.14 moving to Extended Support Release

For an accurate response expressly addressing your situation, we need you to post the SUB-MODEL of your husband's iMac. That is quickly available from "About this Mac..." under the computer's Apple menu. This is an example of the "About.." window from an iMac running macOS Sierra 10.12:

In the example, SUB-MODEL is "(27-inch, Mid 2010")

It is the sub-model that uniquely expresses the highest possible macOS version that a particular iMac can run. It's an important factor in asking here where we can neither see nor touch your computer.

NOTE: for security reasons, please DO NOT post the serial number.

+2. Your husbands computer is technically obsolete meaning it does NOT have the ability to upgrade to current versions of Mac OS. It also means if the computer has a hardware failure that parts are no longer available which means the computer should be replaced.

My advice is to begin planning to replace the computer with a new(er) Mac. The first thing to do is ensure the computer is backed up using Time Machine, if he already is backing up using Time Machine that is great news. Please check Time Machine to verify it is working okay, never assume it is! If he is not familiar with Time Machine please click Back up your Mac with Time Machine. Time Machine requires an external HD to backup to, the one many of us on these forums prefer is the https://eshop.macsales.com/item/OWC/ME3NH7T02/ due to its reasonable price, high reliability rating and the service the company (OWC) provides. A word of warning, operating ANY computer without a backup is shortsighted and almost always results in lost data (photos, financial records, important documents, music, movies, etc.....) so please don't be short sighted!

After you have the backup issue in-check then you can decide on the next Mac. Once you decide on one and have taken delivery you can connect the old Mac's Time Machine backup to the new Mac before turning it on for the first time. When you turn on a Mac for the first time it starts with Setup Assistant, part of the setup is the computer will ask if you are migrating from another Mac. Simply select yes, answer the on-screen questions and the new computer will migrate all of your data from the Time Machine drive to the new computer. This is dead easy and only takes about an hour or two to complete.

denise CLARK wrote:

iMac info: iMac (Retina 5K, 27-inch, 2017)

You can upgrade that Mac as far as Ventura. That would be a jump of six major versions from Sierra.

That Mac is pretty well-equipped – with a nice 27" 5K screen, the ability to accept up to 64 GB of user-installable RAM (not that you're likely to ever need that much), USB-A (USB 3.0) ports, and USB-C (Thunderbolt 3) ports.

Ventura is one of "the most recent three" – and should be for about a year after Sequoia comes out. The version of Safari in Ventura is likely to be recent enough for your husband's bank – and if not, you can download another browser (like Firefox, whose current version will run on anything from Catalina on up).

Both Sierra and High Sierra are obsolete and no longer supported.

Honestly, it’s probably time to consider a new Mac. That one is at least 8 years old.

macOS 12 Monterey is as far as it’s going to be able to go. Monterey is still supported, but will fall out of support later this year when the next major version of macOS is released.

denise CLARK wrote:

When he upgrades to High Sierra, will Safari be automatically upgraded too?

You'll get whatever version of Safari is bundled with High Sierra.

High Sierra is six major releases behind Sonoma. The last point release of High Sierra was 10.3.6, released in July 2018.

You can also try Firefox, which might allow for a more modern browser on your old machine.

Has the OP told us the iMac's model year, yet? There are 2017 iMacs that can run Sierra and Ventura.

You’re right. I should have clarified… IF it’s only 8 years old it can upgrade to Monterey. That was the last model to ship with Sierra.

Either way, I’d say it’s time to look for something new.

Your wording was fine, but based on MacTracker’s database, Ventura is the “best case” OS. That’s all.