Eclipse and Gnome are not much related. Gnome is your Desktop Environment, and Eclipse IDE is just a GUI-based application - it will run on top of any window manager, as long as all the libraries are in place.
For a very light-weight desktop, yet easy to use, you might have a look at Fluxbox. It is small and fast and should be enough for your needs. Other possible options are e.g. XFCE and LXDE.
From the live CD
You seem to be able to get a working connection on the installation media, so here is one idea: Start the arch live CD and setup your network. Then mount your newly installed partition (for example on /mnt) and chroot into your system using
# arch-chroot /mnt
From there, you will be able to update pacman's database and install the desired packages. For broadcom, you will need to install from AUR:
# pacman -Syy base-devel
# pacman -S b43-fwcutter
# curl https://aur.archlinux.org/cgit/aur.git/snapshot/b43-firmware.tar.gz | tar xzf -
# cd b43-firmware
# makepkg --asroot --install
Note: never use --asroot in normal situation.
Without network connection
This is a little bit more tricky here. Compiling from AUR will be harder, so if you can first setup the ethernet using official packages, that will be better. The idea is to let pacman prepare a list of downloads, use another PC and a USB stick to convey the packets to your install. Mount the USB stick on your fresh install and create a list of packages to download.
# cd /mnt/usbstick
# pacman -Sp your_ethernet_driver > pkgs_list.txt
If you really want to install the broadcom drivers (or your ethernet card is also an unofficial packet) also issue
# pacman -Sp base-devel b43-fwcutter >> pkgs_list.txt
Unmount the key and find an internet connection on another PC. Download all the packets using for example curl, wget or simply your browser. If you are really unlucky, the pacman database may be too old and you will not find the packets in their indicated version. You will have to search a little bit round to find the right package. Save all the packets on the stick.
If you go the unofficial way, find the page on the AUR and download the tarball for the packet, but also all dependencies and all sources. For broadcom, for example download the b43-firmware tarball but also the http://www.lwfinger.com/b43-firmware/broadcom-wl-{xyz}.tar.bz2 source tarball.
Go back to your arch and from your stick run
# pacman -U *.pkg.tar.*
For broadcom, (or similar for unofficial packets)
# tar xzf b43-firmware.tar.gz
# cd b43-firmware/
# mv ../broadcom-wl-{xyz}.tar.bz2 .
# makepkg --asroot --install
Note: the third step moves the sources into the build directory so that makepkg finds it locally and do not attempt to download them. And same, do not use --asroot in normal case.
Best Answer
The package you are looking for is called
wpa_supplicant, it handles logging into protected wireless networks.If you use it from Ubuntu (or other debian based distributions) it's fairly easy to set up and the process is rather simple (check the debian wiki for a few pointers). I don't know much about arch linux but it shouldn't differ too much.
If you still want the convenience you know from desktop environments or are for some other reason tied to
network-manageryou can use thecnetworkmanager(website) package that allows you to talk to the network-manager daemon from a terminal.