Thank you for the suggestions.
As I mentioned in my original post, the brcmsmac is already used. Reading
https://wiki.archlinux.org/index.php/Broadcom_wireless [nofollow] the other option would be trying the STA driver (wl).
BackBox offers the installation of this driver, but the installation fails as described in my first post.
I did what you suggested:
sudo modprobe -r b43
sudo modprobe -r brcmsmac
sudo modprobe -r wl
sudo modprobe b43
sudo modprobe brcmsmac
sudo modprobe wl
On the lines referring to the wl driver I get an error, since this is not present on the system. (bcma wasn't present either).
To install the STA driver I ran:
sudo apt-get update
sudo apt-get install linux-headers-$(uname -r) // btw: 2.6.38-14
and then from the menu:
(Start) > System > Additional Drivers
Possibly the install via the menu could have been replaced with this:
sudo apt-get install broadcom-sta-common
sudo apt-get install bcmwl-kernel-source
Then
sudo modprobe -r brcmsmac
sudo modprobe wl
gives "FATAL: Error Inserting wl (/lib/modules/2.6.38-14-generic/updates/dkms/wl.ko): Invalid argument
I blacklisted brcmsmac in /etc/modprobe.d/blacklist.conf and rebooted.
(Start) > System > Additional Drivers says:
Broadcom STA: Activated but not currently in use (

)
Broadcom 802.11n wireless LAN driver: Not activated (that's the brcmsmac, I assume).
It seems that the wl driver is not used, since ifconfig -a shows no wlan0.
To fix the problem I went
sudo apt-get remove --purge linux-backports-modules-*
and restarted.
Finally the wl driver jumped into action.
sudo airmon-ng start eth1 // yes, it's not wlan0
claimed it worked, but
sudo airodump-ng eth1
miserably failed.
Further research shows that the wl driver doesn't support monitor mode.
Conclusion:
The "inbuilt" driver brcmsmac works "a little", sadly airodump-ng doesn't capture any data.
After a lot of hassle installing the wl driver, it turns out that this driver doesn't support monitor mode.
P.S.: This page
http://www.broadcom.com/docs/linux_sta/README.txt [nofollow] claims, that monitor mode is supported by going
$ echo 1 > /proc/brcm_monitor0
Enabling monitor mode will create a 'prism0' network interface. Wireshark and other netwokk tools can use this new prism0 interface.
This doesn't work for me, I get "bash: /proc/brcm_monitor0: No such file or directory".
Strange, the directory exists, but is not writable, not even as root.