With your iMac wireless turned off.. ethernet connected to the TC LAN port..
Do you experience any internet drop outs.. or is this purely wireless?? If both drop out.. we can assume it is WAN connection.. if just wireless drops out we can assume it is purely wireless connection. Both have been reported as going bad.
Is it running 7.7.3 firmware?
Have you tried a full factory reset? If not that is the next step.
The Factory Reset universal
Unplug your TC/AE. Hold in reset. and power the TC/AE back on.. all without releasing reset and keep holding in for about 10sec. The time is not important.. it is the front LED rapid flashing that indicates you are in factory mode.
Release reset.
If it doesn’t flash rapidly you have released reset at some point and try again.
Be Gentle! Feel the switch click on. It has a positive feel.. add no more pressure after that.
TC/AE will reboot after a couple of minutes with default factory settings and will wipe out previous configurations.
No files are deleted on the hard disk.. No reset of the TC deletes files.. to do that you use erase from the airport utility.
When you redo the setup (you will need wireless turned on in the iMac if you turned it off before).. and you should have IPv6 set to link-local only for both wireless and ethernet If you use the later.

Do NOT accept the suggested names for the TC and the wireless. They are WRONG.
Use a name that is short, no spaces and pure alphanumeric. Eg.. TCgen5
What follows is for wireless fixes.. this is pointless if the ethernet is also dropping out..
For wireless I am going to suggest changing to naming both bands. TC24ghz and TC5ghz.. if you end up without problems go back if you want and name it TCwifi.
Use WPA2 Personal security with a pure alphanumeric password.. mixed case and numbers.
You will now need to rediscover the wireless on the clients.. sorry about that.. but that is part of the testing.
Now see if the TC is stable.. no luck.. try fixing the wireless channels.
Click on the wireless tab..

Down the bottom is wireless options.. click that. Here you set the 5ghz name and channels.
Change from auto to 11, 6, 1 for 2.4ghz.. and try setting 40 for 5ghz.. although I am not sure it works.

Tell us how you go.. not much more you can do as Apple does not overwhelm end users with options.. highly underwhelmed really.
If both wireless and ethernet connections drop out.. the issue is the WAN connection on the TC.
Fixing this is much harder.. if the factory reset etc has not done anything tell us.. WAN fixing really needs more testing and there is very little to help you.
If you have available a switch or an older crossover cable you can use them (one or the other or even both) between the modem and the TC.
No luck it is time to take it to the store.