Do we need a ftdi basic board to connect a ftdi 232R USB - TTL serial level converter cable via a 6 pin 0.1" male header into an Arduino pro mini/ nano 328 to upload the sketch?
For the Arduino Nano:
NO, the Nano already includes the USB-to-TTL conversion circuitry. Just use the USB connector as you would with a Uno.
For the Arduino Pro Mini:
YES, you need an external breakout board to perform USB-to-TTL conversion. This board may use the FT232RL chip, but also any other equivalent one like the CH340G. You only need to have the adequate driver in your computer and you'll be set.
If you have "converter cable" you might not need a breakout board because this kind of cable already includes the FT232R (or equivalent). You don't need to use this cable with the Nano.