The 3v3 output pins on the Due to the 5V input pins is fine -- 3.3V will still be recognized as 'high' by the 5V device.
It's going the other way that will cause problems (unless the inputs on the 3v3 device are 5V tolerant, as with the PJRC Teensy 3.1, for example, but that isn't the case for the Due, unfortunately.)
So the only pin in the SPI interface which is an output from the 5V device to the Due is MISO (master input/slave output).
The easiest way to convert this output to a safe level is by a simple voltage divider, e.g. a 18K and 33K resistor in series between the pin and ground, tap the input to the Due from the junction between the resistors.
That's the easy part. But the other gotchas you have to worry about is the values of the pins on both devices while either or both are initializing. You have to ensure that 5V from a connected pin is never going into a 3v3 input on the Due, not as a 'high' output. but also not even as a pullup input. That's the thing that should keep you on your toes/awake at night.
Finally, you'll have to make sure your grounds on both sides are connected at the least, and ideally, you should really be running from a common power supply.
Another thing you could do is to run your pro at 3v3 volts by feeding the Vcc rail with 3.3V directly, rather than feeding "raw" to supply the 5V regulator. Or just get a 3v3 pro mini (would only be running at 8MHz though). Not sure if you actually need 5V, you didn't make this part clear. But running both sides at 3.3V would certainly be a safer option.
Another thing is to have a bidirectional level shifter on all your connected pins. Something like this: http://www.adafruit.com/products/1875 (you may even be able find something cheaper shopping around a bit). That would be also be safer than just using the simple voltage divider setup described above.
Alternatively, you could replace the Due with a Teensy 3.1, and have the 5V tolerant inputs, if that was feasible.
So, several ways to approach this. I'd probably just get a Teensy 3.1 myself, but I'm kind of partial to the Teensies. :-)
You could plug the Teensy 3.1 into something like this if you needed shield compatibility. (I'm partial to these as well, but then of course, I do sell them. ;-)