The A06 BIOS update HERE  is the correct one for a Dimension 4400.  You have to be careful about using one from another model (usually they won't even attempt to try and flash) as any corruption of the BIOS and your motherboard is history and the only way to get it working again is with a new motherboard.

Can you borrow a floppy from someone so you can load the BIOS update, if it is really needed. 
The only reason to update the BIOS is if the list of fixes includes a problem you are having or if it supports new hardware that you plan to install, but considering the age of the Dimension 4400 I doubt that either of these situations apply.

The BIOS is about the last place to look for a connectivity issue. 

Exactly what type of connectivity problem do you have?  If you are so sure that the Ethernet card is the source of the problem, have you tried a new Ethernet card? 

Repost this problem on the Network/Internet/Wireless forum section HERE  so the experts there can help you more.

But, if you RIGHT click on the icon do you get an option to try and fix the connection?  Also, have you tried a different Ethernet cable between the PC and the modem?

Also some of that type of connection problems can be solved by (1) power off the PC (2) repower the broadband modem and wait until it retrains (resync's) and all the lights are green (or whatever is usually normal on that modem) and only after the modem is ready then power up the PC and let it connect. 

The Dell downloads for the 4400 are HERE.  There is a chipset driver (for the motherboard) and several Ethernet drivers for different models.  Have you tried reinstalling the chipset drivers and then the appropriate Ethernet drivers?

Fireberd:  thanks for all the help.  Someone at the the Network/Internet/Wireless forum suggested restoring certain registry entries with an application called Winsockfix, and that did the trick.  I really appreciate the time you took to help me with this!