On a consumer / small business network, you typically have...
#1. Broadband modem. This provides the internet connection to the router.
#2. Router. This provides Network Address Translation so your network can all share the 1 IP given by your ISP, provides DNS forwarding services, firewall etc... In your case an Apple Airport Extreme with 802.11 b/g/n with sumultaneous dual channel N, and gigabit WAN and LAN ports. That should be plenty fast.
#3. Switch. This extends the physical port capacity of the router. FWIW, It is far better to have a single large switch than a stack of small switches from a performance perspective. The uplinks are your bottleneck.
#4. Client devices. NAS, IP Cameras, Phones, TVs, laptops, PCs, Macs etc...
And it should go without saying, but I am going to say it anyway, for your wired ethernet, you obviously need ethernet cables. Since you have Gigabit connections, I am assuming you have category 5e or category 6 cabling for this application.
Obviously something is amiss, and we need to narrow down where the issue is...
When you say network comes to a crawl when doing a transfer from WiFi, do you mean just on the client doing the transfer?
That is to be expected, it's busy, let it do its thing.
Do you mean the wireless network? Again, if you are saturating the bandwidth that is to be expected, there is only so much bandwidth available there, and depending on environmental factors, the advertised speed of your 802.11N hardware may be considerably lower than what you expect. You shouldn't saturate your WiFi with just one big file transfer, but if you are transferring big files, and someone else in the house is streaming Netflix, someone else is Playing Facebook games etc... all on the WiFi, and you've got walls in the way, electrical noise all over the place, steel, copper, and iron all over the place in home construction, well you can see what can hapen to a radio signal... and the bandwidth it carries. If your big file transfers are causing problems for everyone else, you might consider making adjustments in your WiFi driver, depending on make, model, and OS, to throttle your speed back. Say if you are on N600, throttle back to N150 and let everyone else have a crack at the airwaves...
Or do you mean wired clients experience issues when a WiFi client shoves large files over the network? That would be unexpected, and would need to be dug into further.