First of all, give your network a unique SSID. Do not use "linksys". If you are using "linksys" you may be trying to connect to your neighbor's router. Also set "SSID Broadcast" to "enabled". This will help your computer find and lock on to your router's signal.
Next, temporarily turn off your computer's software firewall, including Windows Firewall, and see if that helps.
Also, what "security code" are you using? For WEP, enter WEP "key 1" (found in the router) into your computer, not the WEP password or passphrase. For WPA or WPA2, enter the password or passphrase into your computer.
Also, in the computer, go to your wireless software, and go to "Preferred Networks" (sometimes called "Profiles" ). There are probably a few networks listed. Delete any network named "linksys". Also delete any network that you do not recognize, or that you no longer use. Delete your current network (this will clear any old settings.) Reboot computer. Return to "Preferred Networks" and re-enter your current network info (SSID, encryption (if any), and key (if any) ). Then select your current network and make it your default network, and set it to automatic login. You may need to go to "settings" to do this, or you may need to right click on your network and select "Properties" or "settings". Reboot computer. It should connect automatically to your router.
If the above does not fix your problem, download and install the latest driver for your wireless card.
I think you can try to update the firmware of the router, to check if that stops connection dropping. To update the firmware of router try this: 1. Go to www.linksys.com/download and select correct model number and version number. Download the firmware and save it on your computer. 2. Check the file it sould be firmware image file like .bin etc. If it is a compressed zip file extract it to get firmware image file. 3. Open setup page of router by typing http://192.168.1.1 in browser’s address bar, and typing password as "admin" and no user name. 4. Go to administartion tab there 5. Go to fimware update sub-tab there 6. Browse for the firmware image file and click update 7. when update completes turn off modem & router. 8. After 2 minutes turn on modem and after 2 more minutes turn on router. To get network key, connect your computer to one of the four ports on router using ethernet cable. 1. Open setup page of router by typing http://192.168.1.1 in browser’s address bar, and typing password as "admin" and no user name. Use your password if you have changed it earlier 2. Go to wireless tab. 3. Go to wireless security sub-tab 4. If security mode is wpa or wpa2 note down passphrase/shared key, if security mode is WEP note down wep key number 1. This will be your network key Well you need to input all the wireless settings of your wireless network on Wii. To get these settings: 1. Open setup page of router by typing http://192.168.1.1 in browser’s address bar, and typing password as "admin" and no user name. 2. Goto wireless tab. Here you will get the wireless network name/SSID. Here make sure wireless SSID broadcast is enabled. 3. Now goto wireless security sub-tab and here you will get the security mode and the security key. If security type is WPA note passphrase/shared key, and if its WEP note down wep encryption level and key 1 from listed 4 keys