After several hours of trouble shooting, I finally got the BeagleBone and a LS20031 gps to talk.
I hooked the LS20031 to UART5 of the BeagleBone. The LS20031 has 3.3V TTL serial, so a level converter was not required. UART5 looks to be the toughest of the serial ports to use. The trick to getting UART5 RX and TX pins (37 and 38) to work, is to set them to MUX mode 4. Most of the other serial ports work in mode 0.
A second problem is the LS20031 is really picky about startup. If you have the RX or TX pins connected and power cycle, the unit does not always come back online. This is indicated by the red LED being constantly on. The only way I have found to make it power on correctly is to reset the BeagleBone. Note: I am using the 3.3V power supply from the ‘Bone.