Display Ports

Please refer to Use your iMac as a display with target display mode. It has been archived, so its information may no longer be applicable.

There are hardware and software requirements: "Make sure that your iMac is using macOS High Sierra 10.13.6 or earlier. You can't use target display mode with later versions of macOS, or with Boot Camp and Windows."

Have you verified that your iMac is one of the specifically supported size-year-models here:

https://support.apple.com/en-us/HT204592

If so, which one are you using specifically?

Did you boot up the iMac and toggle on TDM mode by pressing Command-F2 or Command-Fn-F2?

Does your MacBook Air in About This Mac > System Report under Graphics/Displays show the additional display there at all in additional to the built-in display?

If (big if) all of that is true, does System Preferences > Displays show you an "Arrangement" tab and a "Detect Displays" button?

Hmm, I don't believe you have to have an actual miniDisplayPort cable and that a full Thunderbolt cable should work fine.

Is the iMac actually switching into Target Display Mode?

What's the model year and screen size of your iMac?

If there's any doubt about the functionality of your Thunderbolt cable to Thunderbolt ports you should put one Mac into the similarly named (but totally different) Target Disk Mode and verify that works:

https://support.apple.com/guide/mac-help/transfer-files-mac-computers-target-disk-mode-mchlp1443/mac

Target Display Mode (turning your iMac into a monitor-only mode) is extremely specific.

In general, only older 27-inch iMacs support this. And only those Mid-2014 or older support this.

Since you are running 10.12 Sierra, if you do have one of those 27-inch iMacs that have this feature it would work.

If you aren't aware, then only way specifically to drop your macOS is to do a Shift Option Command R recovery to get to the oldest macOS (or OS X) your hardware supports, then upgrade only to 10.13 High Sierra and stop there. You'll want to upgrade to 10.11 El Capitan first from whatever old version you get (10.7, or 10.8 probably), then to 10.13.

Ok, so for your iMac it does turn out that it DOES require thunderbolt cable (unlike some of the earlier iMacs). And you say have one (or rather two!) so that's great. Make sure your cable has the lightning logo on the ends.

Explanation: Target Display Mode isn't literally turning your iMac into a generic external display that you could hook up any Windows computer or game console to. That's what the earlier models could do...running Xbox on an iMac screen was awesome, btw.

For your iMac Late 2014 though, target display mode acts like thunderbolt accessory device (i.e. requiring drivers and specific support...which of course the MacBook Air running Sierra has built-in.

In other words, if it can't do a data connection over thunderbolt it's not going to work. How do we know a data connection over thunderbolt is working or not? Test Target Disk Mode.

Your computer most likely has a thunderbolt port, the plug is the same as mini-DP. So you’d need a thunderbolt 1/2 cable.