I am running Linux Mint 17.1 on a 32 bit laptop, Fujitsu S6240.

The built in wireless adapter, is, I believe a Broadcom BC5751M on the PCI buss. There is an external switch that allows one to turn it on or off.

I am attempting to use a ThinkPenguin N150USB USB wireless adapter. If I turn off the internal wireless with the switch, and insert the usb adapter it does not show up on the list of available connections. In fact, there is a red X for wired connections. If I then turn on the internal wireless adapter, I see two listings for most of the available networks, one associated with eth1 and one associated with wlan1.

I can connect to either one. If I remove the usb card, only the eth1 networks remain.

Again, with the internal wireless adapter's internal switch off, I cannot see any available networks. This is true if I boot with the usb adapter inserted or if I insert it after booting. If the internal wireless switch is set to off, the external wireless doesn't work.

If I am using the external adapter, I would like to turn off the internal wireless in order to minimize the battery drain. Any suggestions?

There's an internal thread on the topic here at LQ

blacklist the broadcom driver aka kernel module.
a quick search will tell you how to do that for your distro.
after that reboot, what's the situation?