Out of the box the Broadcom 4312 doesn't work on LMDE nor  most of the distros out there. With LM or Ubuntu you can use jockey to install the drivers with Debain it isn't hard it just takes a few more steps. So now we'll see if we can get your Broadcom 4312 wifi working.

I have several laptops with this card in them so what I did was write a script to automate the install. What I am going to so here is post the script with the comments so you can either cut & paste the commands into a script or type them in, your choice. 

# Indicats a Comment

Bold - Commands to type into the terminal if not used in a script

All this will be done in a terminal so let's get started

Menu/Terminal - this will open the terminal window

I have found I have much better luck if I make sure my system is up to date before trying it install the Broadcom driver so lets do that first

sudo apt-get update

sudo apt-get dist-upgrade

# ****************** Script Start Here ***************************************

#! /bin/bash
# Install the wifi drivers for Broadcom 4312 wifi
# Add contrib non-free to the app source list befor using - Note this may not
# be necessary on LMDE
############################################################################

## Update the list of available packages. Install the module-assistant and
## wireless-tools packages:

sudo aptitude update
sudo aptitude install module-assistant wireless-tools

## Build and install a broadcom-sta-modules-* package for your system,
## using Module-Assistant:

sudo m-a a-i broadcom-sta

## Rebuild your initial ramdisk, to blacklist modules defined at
## /etc/modprobe.d/broadcom-sta-common.conf within initramfs:

sudo update-initramfs -u -k $(uname -r)

## Unload conflicting modules:

sudo modprobe -r b44 b43 b43legacy ssb

## Load the wl module:

sudo modprobe wl

## Verify your device has an available interface:

 sudo ifconfig

## Configure your wireless interface as appropriate.

##Now so the driver will load when the computer is booted 

sudo gedit /etc/modules

add wl to the end of the file ending with a C/R

## ************ Script ends here ***********************

Now all that is left is to start network manager and configure it for your wireless

Click the Network applet in the lower left hand corner (this assumes that the panel is in it's original position) you should now see you network listed. Click on it and enter you password if needed.

If you connect you can now unplug your ethernet cable.

I don't know if it is really necessary but I normally restart at this point so I can make sure it is all working whiile it is still fresh in my mind as to what I am doing.

This tutorial may become out date as I believe that  Clem and the gang are working on a jockey type program for LMDE but until them I hope this helps to get the BM4312 working

Your milage may vary

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Through the years I have had so many issues with this chip set that I will no longer buy a laptop with this chipset. Had one laptop that worked, did an update and it wouldn't work so I reverted back, still wouldn't work. Finally bought an Intel wifi card.