How can I disable the Internet on my DAW in a network of 3 computers?

Gerry, what operating system are you running ? Are you using a router ? Is your LAN wireless (wi-fi) ?

You should be able to just click on the network connections icon in your taskbar, then right click over your network name and select "disconnect".

Is your DAW computer a hub for your other computers ?

More info about your setup would be useful.

won't that disconnect me from the network in addition to disabling the internet?

Yes it is just like unplugging the cable. 

The most straighforward way to do this on a single machine is to make it so that the machine does not see the cable modem which is probably serving up a temporary IP addres to your router which is acting as a DHCP server to give your local machine an IP address. The answer is to set up the machine with a static IP address and disable the local gateway and DHCP service. 

In the start buton search box type cmd.
At the command prompt in the window that appears type "ipconfig" no quotes. The response gives you the current IP (IPV4) address, subnet mask and default gateway currently assigned to your computer . Write these down,and exit the command window.

Type "network connection" into the start menu search box. 
Click the properties button in the next window. 
Double click local area connection in the window that appears. 
Double-click Network Connections, and then Local Area Connection. 
In the Local Area Connection Status dialog box that appears, click Properties.
Click to highlight the words Internet Protocol [TCP/IP] (do not remove the check in the check box), and then click the Properties button.
Select the radio button for Use the following IP Address, and enter the IP address of your computer. It might be wise to change the last three digits of that code to something lager than the number of computers on your network to prevent a conflict if you try to connect another computer automaticlly. So if your computer came up in ipconfig as 192.168.0.100 changing it to 192.168.0.120.
Enter the code that you wrote down for the "Subnet Mask" field.
Click the "use the following DNS server button".
Make the "Default Gateway" field, empty by deleting anything that is there.
Make the "Preferred DNS Server" field and the "Alternate DNS Server" field, blank by deleting the information that is there.

You can achieve a more nuanced effect by blocking the internet via the firewall in Windows or the one in your router. So if you wanted to leave access available to the Windows update site for example  but block everything else that is possible, but more complicated.

  Thanks Slartabartfast. I woudl like a little more clarification. So if I put all the numbers in from the ipconfig, does that mean that the internet will work fine for my DAW? And further more if that's true, then if I change the IP address number like you sugggested and left the default gateway blank, then that would effectively disable the internet from my DAW and still allow me to be connected to my other computers in the network? If that's all true, then what if I instead of totally blanking out the default gateway, what if I just change a number. Would that give the same result? The reason I'm asking is I'm trying to come up with the closest thing I can to a toggle on and off of the interent. If all I'd need to do is create a shortcut so I can get to that box quickly and just change 2 numbers, that's close enough to a toggle. I may even create a macro that does that. Maybe even running a script could accomplish all that in one swoop. Thanks for your excellent advice.

So if I put all the numbers in from the ipconfig, does that mean that the internet will work fine for my DAW?

OK so I was assuming you are running Windows 7. If not then the settings are in slightly different locations. It was also not clear that you wanted to switch back and forth rapidly. 

You do not have to put any numbers into the boxes if you are using a router that acts as a DHCP server in order to get back to the internet. Just click the radio buttons to obtain IP address and gateway automatically and reboot the computer or type "ipconfig /renew" at a command prompt and your router should download those numbers to your computer.  If you type in a static IP address without knowing the addresses assigned to your other computers you risk conflict. That is avoided by letting the router do the assignment (automatically assign radio button) or using an address higher than the router will automatically assign.

The important thing is that if your computer knows (because you type it in) or can automatically discover (by interrogating your router) the gateway address it can get to the internet. If the gateway address  is not present AND you have told the computer that it should only look for that address at a blank (or otherwise incorrect)  address, it will not find the gateway and cannot see the internet. 

If you are looking for a quick way to switch between internet accessible and not you can use netsh.exe scripts from the command prompt or in bat files.

http://www.petri.co.il/configure_tcp_ip_from_cmd.htm

http://windowsitpro.com/windows/use-netsh-easily-change-ip-addresses