CTS and GND are shorted on the MiniPro IIRC - In fact GND is black, CTS is the second pin. Having CTS grounded means the computer can always send serial even if nothing's listening.My Sparkfun 5V Basic breakout has CTS going to the chip, not ground. Perhaps there is a different design or yours is shorted - it should still work if CTS is grounded as that's how it would be if plugged into the Mini Pro anyway.
I tried that but it didn't seem to work. I've ordered a new Arduino Mini Pro and one of the newer 3.3V FTDI basic breakout boards so hopefully this time round it'll work if I avoid scrubbing it with the abrasive fiberglass pen. I'll clean it with isopropyl alcohol instead I think.
I can't get the things to work so I'm wondering whether my soldering job may be the problem!
I've still got a lot to learn!