ETXuser I connected AsiAir plus to the mount HTX jack with an Ethernet cable.
I owned a small ETX some 20 years ago, and don't remember that it supported Ethernet.
Are you sure that the RJ45 connector is not the one used to connect to the handbox? Many mount handboxes (including my RainbowAstro mount) use an RJ-45 connector, just because the cables are easy to obtain. But they do not carry Ethernet signals, and should never be used for that purpose, in case you burn something up.
If memory serves, my old ETX had a second smaller RJ connector, perhaps an RJ-11 (?) for serial port for connection to a computer.
If so, you may need to go from that RJ-11 port to a regular 9 pin serial port (I think Meade sells a cable to do that and it may have come in the ETX box), and then from the 9 pin serial port to USB with yet another adapter (you can easily find 9 pin serial to USB adapters). With the ASIAIR, the FTDI USB cables will give you less problem than Prolific USB cables. There are a couple of other threads on this forum where people are still having problems with Prolific drivers on the ASIAIR.
You can check the ETX manual on how to connect the mount to a computer's serial port.
You might find further clues by searching the web for keywords combinations like ETX, serial, DB-9, USB, FTDI, Virtual COM port, etc.
Good luck. And again, do not power the ASIAIR on with its LAN port connected to a connector at the mount that is meant for a handbox. You can connect the ASIAIR to your router to test if its LAN (Ethernet) port is still functional, in case you need to use Ethernet in the future.
Chen