Open a Terminal and install the bcmwl-kernel-source package:

sudo apt-get update
sudo apt-get --reinstall install bcmwl-kernel-source

Note: If you see the message "Module build for the currently running kernel was skipped since the kernel source for this kernel does not seem to be installed" then you are missing the appropriate generic linux-header package(s).

To test the driver (and remove the need for a computer restart) use:

sudo modprobe -r b43 ssb wl brcmfmac brcmsmac bcma
sudo modprobe wl