How to look at Airport logs

Save configuration and replace it with another one from those that have their radios disabled. Maybe this will be enough of a solution.

Why would this be happening? I don't know but it's either power supply or its OS crashing. Or both.

From the Apple Support Airport Express FAQ:

Question: Since AirPort Express only has one Ethernet port, does it act as a LAN port, WAN port, or both?

Answer: Depending on how you have it set up, the Ethernet port can function as a LAN port (defined as either simple bridging or sharing your Internet connection via network address translation) or a WAN port (connecting to your broadband Internet service provider or upstream router). However, it never acts as both simultaneously. By default the AirPort Express Ethernet port acts as a WAN port. When it is set up as a secondary extended Wi-Fi base station, it can act as a LAN port. Assuming that your Internet service provider only gives you one IP address (the most common scenario for homes), this means that AirPort Express can only accommodate wired clients when acting as an extender or bridge. Therefore, if you need to support wired clients and only want to get one base station (or router, as it's the routing capacity of AirPort Express that is relevant here), you should get the AirPort Extreme Base Station or Time Capsule instead.

According to an answer to a question on an Apple Store page:

Yes, certainly. Connect Ethernet cable from the Express to your non-wireless computer which meets the above listed "System Requirements" and use AirPort Utility to direct the Express to "join" your wireless network from the list which will show on the popup presented to you. The Express will then function as a wireless adapter (receiving signals) for your computer, but in that configuration it will not extend the wireless network to other devices.

You can accomplish this task via the Airport Utility. In one Apple discussion forum discussion, a user (Bob Timmons) notes that:

new AirPort Express 802.11n can be configured to "extend a wireless network". That is a much simpler setup with much higher performance than WDS can offer.

You'll need to use Apple's Airport Utility to set up your Airport Express. Two approaches that you can take are WDS or Extended wireless network, as noted in this Apple Support KB.