This forum post seems to have helped most people facing this issue.
I have reproduced the necessary steps here for the sake of simplicity.
In the Additional Drivers application, check for any drivers pertaining to
STA and disable them.
Open Terminal and enter the following command:
sudo apt-get remove b43-fwcutter broadcom-sta-common broadcom-sta-source firmware-b43-installer firmware-b43-lpphy-installer firmware-b43legacy-installer bcmwl-kernel-source
Now, enter the following commands:
echo "blacklist bcma" | sudo tee -a /etc/modprobe.d/blacklist.conf
echo "blacklist brcm80211" | sudo tee -a /etc/modprobe.d/blacklist.conf
Open Terminal again and enter the following command:
sudo apt-get install bcmwl-kernel-source broadcom-sta-common broadcom-sta-source firmware-b43-installer
After the package installation is complete, reboot.
Open Additional Drivers.
Broadcom STA driver.
Your Wifi should now be active and functional.
This seems to be a documented bug.