To check for software updates: (Apple menu) > Software Update... Whatever is offered is installed on your Mac.
When you launch AirPort Utility it will inform you of firmware updates for your AirPort Base Station, if any exist. Then are installed on the AirPort Base Station.
Neither update is likely to address your wireless connection problems, which require more information to diagnose. Substitute your Linksys and if the problems go away, you may have a defective AirPort Base Station.
My Linkskys quit working finally is why I got the apple. And Linskys has the most annoying customer service ever!!!!!! I have read that a lot of problems is probably mine & my fiance's macbook pros. That there has been a lot of problems/concerns/discussions/frustrations with the wifi being dropped a lot. When I got to my network preferences or the "signal icon/ upside down triangular thing" at the top of my screen it says "Wi-Fi: looking for network" alot. I even have it under advanced to remember my network. Maybe this is the main problem. And it seems Apple is doing nothing about it. My PC works fine under wifi. I just would rather use my Mac that has most of my important stuff on it. But the losing wifi is EXTRA FRUSTRATING!!! Especially when we pay so much $$$$$$$ money for a Mac compared to a PC. Oh well. I do appreciate your time of trying to help me.
Do a full factory reset on the AE.
Setup again and use all short names, no spaces and pure alphanumeric.
If there are wireless issues, IMHO it is better to use different names for 2.4ghz and 5ghz at least until you sort where the issues are.
Use fixed wireless channels.. for 2.4ghz there are only 3 non-overlapping channels 1, 6, 11 (or 13 in ETSI regions). Try each in turn. 5ghz is less of an issue, Try low down,, 36-40 and one of the higher channels.. >150.
Use WPA2 Personal security with password, 10-20 characters enough except for the most paranoid.. pure alphanumeric.
For each laptop clear out all the old networks.. totally empty the retained networks. Then discover the wireless network again and set it up.. that is enough to fix it for most.
See Tesserax doco on issues with sleep .. these can also help with your dropping out.
But you need to be in areas of the house that get at least reasonable signal and not too much interference. If you are in a heavily used wireless area.. check what is available in terms of free channels or force it to 5ghz.
I agree with John.. the latest firmware is not a fix.. indeed people generally report more issues.. 7.6.3 is not great.. stick to 7.6.1 for now.
There may be competing wireless networks in your area that are preventing reliable communication. Is that a possibility? When the Wi-Fi network becomes saturated you may not be able to enjoy a reliable connection even if your Mac is near the Extreme.
The only way to mitigate that annoyance is to constrain your Macs to use the Extreme's 5 GHz network. Give it a unique name as in the following screenshot:
Then, make sure your Mac connects to the 5 GHz network, and delete any other ones in your Preferred Networks list. 5 GHz will practically eliminate any concern about interference, at the possible cost of reduced wireless range. Experiment with this and determine if it helps. Your PCs and older devices can continue to use the 2.4 GHz network since the Extreme creates both of them simultaneously.
If all else fails reconfigure your Extreme with a "hard reset": press and hold its reset button continuously, five to ten seconds or so, until its LED flashes rapidly. After it starts up again, reconfigure it with AirPort Utility. It is possible for an internal parameter to become corrupted that only a "hard reset" can fix. After that, all you can do is take it to Apple for evaluation if you suspect it's not working properly.
I agree Linksys's customer service fell into an abyss after Cisco's acquisition of the company.