Backup your data before installing anything new.
First go to Apple menu -> About This Mac.
If it only says Mac OS X 10.7, you can't get the latest browser for 10.7. You'll need to install the 10.7.5 combo update from here:
OS X Lion Update 10.7.5 (Client Combo)
Then once you verify it installed successfully through Apple menu -> About This Mac, go to Apple menu -> Software Update. That will let you run the security updates which will bring you to Safari 6.1.6.
Mind you, you'll also need the latest version of Microsoft Silverlight
https://www.microsoft.com/getsilverlight/Get-Started/Install/Default.aspx
You may find a different browser with Silverlight is less trouble than having to upgrade your Safari.