How to update a MacBook Air OS X 10.6.5, to High Sierra?

You will have trouble connecting to any https Web site when running Snow Leopard. The original version of https security was found to be less secure than desired, and there was a movement to upgrade browsers and Web sites to use stronger security protocols. I believe that the version of Safari in Mac OS X 10.11.6 (El Capitan) has at least the first, major upgrade – but that the versions of Safari in Snow Leopard definitely do not.

Assuming that you have a working Snow Leopard system, I believe that you will need to upgrade to High Safari in stages.

1. Using a modern Mac or PC, download the El Capitan installer or the Sierra installer onto a USB flash drive. This will not create a bootable installer – just a USB flash drive with a .DMG data file on it.
2. Transfer the USB flash drive to the old Mac and copy the .DMG file to the old Mac.
3. Mount the .DMG image by opening (double-clicking) the .DMG file. Run the El Capitan or Sierra installer which you will find inside.
4. After you have successfully upgraded to El Capitan or Sierra, you may want to make a bootable clone backup of that system, using Carbon Copy Cloner or SuperDuper!
5. Next, using the old Mac, booted under El Capitan or Sierra, use the High Sierra link to download and install High Sierra. The High Sierra installer is in the Mac App Store – and the versions of Safari and the App Store that are in El Capitan or Sierra should be just new enough to let you get it.

How to download and install macOS - Apple Support

Once you have High Sierra running, you may want to install iTunes 12.8.3 and Firefox 115 ESR. Firefox 115 has not gotten new features in a while – but it is newer than the version of Safari in High Sierra, and it will still get "critical security updates" through March 2025. (That's a slight reprieve. Mozilla had planned to end them by now.)

iTunes for Mac Manuals and Downloads - Apple Support

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