Yep, according to the schematic of the Sparkfun FTDI Basic 3.3V board, the 3.3/5V jumper option to the “power” output also supplies VCCIO, so that is a problem to solve. A simple resistive voltage divider (e.g. 22K + 33K) from TX on the FTDI to RX on the arduino, and a direct connection from TX (arduino) to RX on the FTDI board will allow you to safely use the 5V IO option. Google “3.3V level convertor circuits” for more options.
Another possibility is to disconnect VCCIO from “power” and reconnect it to the built in 3.3 V regulator while using 5V for power output. You might ask SF about how to do that, or inspect the board closely to see if cutting a trace and rewiring will work. I use Pololu’s version of the board which gives more options.