How can I share screen from MacBook Air to 2007 iMac using a wired connection?

The 2007 iMac is too old, falling outside the very narrow list of iMac models that could do Target Display Mode (TDM) in conjunction with a supported Mac and macOS.

You Macbook Air would also have to be running macOS 10.15 or older to use a TDM-capable iMac as a display.

Network screen sharing is best accomplished with a wired ethernet connection, preferably through an Ethernet hub to avoid the sticky issue of whether or not a 2007 iMac requires a cross-over ethernet cable.

Note that network screen sharing will likely be laggy, and cannot do extended desktop.

srspicer wrote:

My NB Air does not have a Mini Display Port, but it does have a thunderbolt port. Looks like I need a Thunderbolt to Mini Display Port cable. I don't know if there is such a beast, I'll look into it. Hopefully my local Apple store has something.

Your 2017 MacBook Air has a Thunderbolt 2 port, with a Mini DisplayPort, that can act either as a Thunderbolt 2 port or as a Mini DisplayPort, depending upon what you plug into it.

It "supports up to 3840-by-2160 resolution at 60Hz on an external display … at millions of colors." (I don't know what performance would be like driving a 4K display, given that we're talking about an Intel integrated GPU on an old mobile Intel processor.)

So there are many standalone monitors that you could attach easily using

• A Mini DisplayPort to Mini DisplayPort cable
• A Mini DisplayPort to DisplayPort cable
• A Mini DisplayPort to HDMI adapter

But you can't use your 2007 iMac as a monitor. It's not designed to accept video input.

MacBook Air (13-inch, 2017) - Technical Specifications - Apple Support

srspicer wrote:

The ports on the 2007 iMac are as follows; 3 USB ports, 1 Firewire 400, 1 Firewire 800, 1 mini Display port and 1 Ethernet .

According to MacTracker, none of the 2007 iMacs have Mini DisplayPort. They all have mini-DVI, a different kind of video connector.

I'll try to do something with cobbling together a cable, if I can get from thunderbolt to mini display port. ???

Don't bother. Your iMac is not capable of being a display – so getting a cable wouldn't help, even if you got a cable that would work with a standalone monitor that had the same type of video connector.

Franob wrote:

Use your iMac as a display with target display mode - Apple Support

Neither of the OP's Macs support Target Display Mode. The 2007 iMac is too old. The 2017 machine is a MacBook Air, and MacBook Airs never had the ability to act as Target Displays.

You don't. Target Display Mode was deprecated years ago.

Use your iMac as a display with target display mode - Apple Support

srspicer wrote:

I'll try to do something with cobbling together a cable, if I can get from thunderbolt to mini display port. ???

Don't bother. It won't work.