A failing base station usually doesn't fail to do certain things-it fails to do anything. You'll see the lights cycle through something that looks sort of like the start up sequence, but it goes on and on. You will probably also hear a sound coming from the device. This sound could be a high pitched whine, a popping or cracking sound, even a sound like crumpling paper or plastic.
The sound is the key to the problem, as it comes from the two capacitors that are the heart of the power supply. Apple went for asthetics rather than robust electronics in designing the original power supply, so the capacitors are "under powered" for the application. Eventually, depending on conditions and usage, they can fail. When they fail they arc internally-the sound you hear is the sparks jumping through parts of the capacitors.
I hope that helps calm your fears. You are probably suffering from a problem at your ISP, or with your ISP's DNS server, not with your base station.