For the record, if anyone else is curious, I got Kismet working with my Broadcom adapter. I needed to
sudo get-apt install kismet
sudo gedit /etc/kismet/kismet.conf
Then I had to edit kismet.conf to include the following line:
source=bcm43xx,eth1,Broadcom
and update the line that says suiduser. After that just running kismet worked fine. Hope that helps anyone else with the same problem I had.