The hardware is Broadcom BCM4312. I briefly had Mint LDME on this little beast, and using that I got wireless working by following debian.org's instructions for Wheezy. But of course Bodhi is not based on Debian and I suppose I therefore can't do that. (Still, it does go to show that wireless doesn't need any esoteric firmware from Dell.)
I looked for Bodhi-specific advice and at first didn't find any, so then looked for Ubuntu 11.10-specific advice. The author of this page says he got his (different model) Dell to work by simply (1) getting rid of the unwanted
sudo apt-get purge broadcom-sta-common broadcom-sta-source bcmwl-kernel-source
(Done!) and then (2) installing what is needed:
sudo apt-get install b43-fwcutter firmware-b43-installer
b43-fwcutter was already installed, but I can't find firmware-b43-installer. (I looked with Synaptic too.) I mean, I don't know where I can download it from.
Now that I look around this forum a second time, I do see the thread Continued Broadcom wireless woes. But this is different -- he has a troublesome connection with Broadcom BCM4312, whereas I have no connection.
Or anyway, at the top centre of my screen I see a little icon of two screens with a red cross next to one. Right-clicking it brings an option "Edit Connections". Following that with "Wireless" brings a blank. I presume that I should see a list of SSIDs there but I'm new to Bodhi and Enlightenment and for all I know I've got the BCM4312-specific stuff installed properly and am missing some other essential part of the process of getting wirelessly connected.