I really didn't want to bump this thread, but something new has come up.
I did some tests after installing the firewall, checking to see if the old computer could connect to the network. In the basement it was a total failure. However, when I connected the computer to the router in the library, the computer did connect to the network and I was able to get internet access, plus I was able to get a deathmatch going in DOOM95, much to my satisfaction.
However, as the computer is supposed to remain in the basement, I have come to the conclusion that the Ethernet LAN card isn't worth it. This is a wired one, and I went ahead and purchased an ISA wireless card. The card uses the IEEE 802.11b specification along the 2.4Ghz band, which is what the router in the library supports along with its main specification which is 802.11g. As a result of this, the first ethernet card will get a removal when the time comes. The card is new with the discs and instructions supplied.
If all else fails after getting the wireless card, the only connection this old computer will use is null-modem. That shouldn't happen, as I checked and one of the wireless cards I've successfully used with the Dell in the past used the IEEE 802.11b specification along the 2.4Ghz band. I expect no issues. And with the wireless extender in the basement, if it does connect, the signal will be strong. I won't be adding any more cards to that old computer after this.