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You need to upgrade the Safari browser to latest version.
What version of Mac OS X are you using ? Click on the Apple icon and select About This Mac. You can find the version number there. It is like 10.6, 10.7 etc
Ok, I will send you instructions in here to solve the problem.
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You have two options in here. Either you have to upgrade your Mac OS X 10.7.5 to latest 10.11 El Capitan to get latest Safari or you have to use different browsers like Chrome or Firefox.
To upgrade to latest version, go to Applications folder and open App Store application (or click on Apple menu and select App Store). In App Store, click on Updates tab and do all updates.
Then click on Featured tab and you will find OS X El Capitan there. Click Get to download. Once it is downloaded, it will ask you to install it. So install. This will upgrade both your OS X and Safari to latest version which will definitely fix the problem.
If you don't want to upgrade, then you have to download another browser like Google Chrome or Mozilla Firefox and access the banking website from those browsers.
You can download Chrome from here --> https://www.google.com/chrome/browser/desktop/
or download Firefox from here --> https://www.mozilla.org/en-US/firefox/new/
You should be able to access your banking website in Chrome or Firefox.
Please let me know if you have any difficulty and I will walk you through the steps to install them.
You can also download El Capitan from here --> https://itunes.apple.com/in/app/os-x-el-capitan/id1018109117?mt=12
The OS X El Capitan is a 6GB file and it woud take a few hours to download and install it. You have to upgrade to El Capitan to get latest Safari. If you don't want to upgrade the OS X, then you can use another browser like Google Chrome or Firefox.
It will only take a few minutes to download and install Chrome or Firefox on your Mac. You will be able to access the banking website from Chrome or Firefox.