I cannot download airport utility. It says my system does not support it. I have 10.10.5

You can't. It is installed with OS X into the Utilities folder. Why do you need to download it?

You don't need to download AirPort Utility......it is already installed on your Mac as part of the Mac operating system.

On your Mac......open Finder > Applications > Utilities > AirPort Utility......and you are all set.

Apple ceased support of all models prior to N wireless.

They changed the airport utility to prevent you loading it at Lion.. so that is already several OS versions that do not support your older hardware.

You can use the version on windows.. or get a Mac running older OS.. or simply load old OS onto external drive in your Mac and boot from it.. easy peasy really.. then you can configure your old extreme or express to your hearts content.

Or google for Yosemite 5.6.1 airport utility.. there is a way to install it.. considered very naughty around here.. so no links allowed.. google therefore is your friend.

" This version of Airport Utility does not support this base station."

I have an Airport Extreme. When I try to download the latest updated version it does not allow me to open it as it

say my system does not support it.

This would indicate that you have an older version of the AirPort Extreme which is no longer supported by Apple.

You still might be able to use the older AirPort Extreme if you have....or can borrow for a few minutes.....a Mac that is running an older operating system like Leopard (10.5.x) or Snow Leopard (10.6.x). A PC with AirPort Utility installed in the device might work as well.

I'm not recommending that you try to install unsupported software from the Internet, but that option does exist if you are willing to accept the risk and change some of the default security settings on your Mac. We can't provide any help for something like that on this forum, but it's your call.