All non-vintage MacBook Air support target disk mode, including the Apple Silicon based MacBook Air released in 2020.
Many vintage and obsolete Air still support target disk mode as well.
Target Disk Mode is not supported on older Macbook Air that lack a Thunderbolt port. The Mini DisplayPort is purely a video out port, and cannot be used for data transfer, despite the identical connector. Since thunderbolt is far faster than gigabit ethernet, it’s preferred due to speed as mentioned in the comments below.
Any MacBook Air that have Thunderbolt 3 or 2 support target disk mode.
For all devices - regardless of ability to boot to target disk mode, an additional supported option for transferring data between two Macs is using a wired connection and Migration Assistant. Since MacBook Air have no ethernet ports, you will need to purchase the Apple USB to Ethernet Adapter for the older model MacBook Air which does not have a Thunderbolt port, an Apple Thunderbolt to Ethernet Adapter for the newer model which supports Thunderbolt or one of the many third party USB adapters.
If you do not want to purchase the adapters, you could migrate the data off the Air using file sharing or have the Mac opened by a service technician to remove the storage and assist with a converter or another Air to house the drive while it's being accessed.
By deleting all files that are non-essential you can speed up the time of transfer, but the USB to ethernet adapter is 100MB speed and likely the fastest you could transfer files on your Air from 2014 or before. Since 2014, much faster WiFi and Ethernet adapters mean faster transfer speeds than before.