Your driver is a "common anode" type driver.
Means, that the controller (in your case you will be using the Arduino for that) will switch ON/OFF the optocouplers of your driver to control it, thus
- ENABLING your driver, so that can listen to the Arduino's commands (STEP/DIR)
- controlling the direction of the motor turn (left/right) by switching the DIR input HIGH or LOW
- sending STEP pulses to make the motor move at a certain speed, depending on the frequency of the pulses being sent
Before you continue reading -> 2 warnings:
a) get acquainted to setting the stepper motor's max. current (have a look at the data sheet of your motor and the instructions of your driver, how to limit the motor's coil current) and do this before starting any testing.
b) NEVER connect/disconnect your motor when power is ON (regardless if it is turning or not).
If you don't obey these two rules you might fry your motor or kill your driver.
Let's go on with something that is not that frightening 
Wiring:
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you don't need addt'l transistors as I assume that the Arduino is able to switch the optocouplers ON/OFF without any further assistance by transistors - but pls check, what the datasheet says, how much current is needed by your driver to switch an optocoupler on -if that value exceeds 20mA, then come back and we have to talk about transistors indeed
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connect the NEGATIVE inputs "PUL", "DIR" and "ERC" with any digital I/Os of your Arduino. (e.g.:
PUL- <-> Due "8"
DIR- <-> Due "9"
ERC- <-> Due "10"
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connect Arduino's Vcc (5V-Pin) with "PLS+" AND "DIR+" AND "ENA+"
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DON'T CONNECT the GNDs !!!!!!! of the Arduino and driver !!!!!!
Code:
You can use / test with any code which is able to drive a bipolar stepper through a dedicated driver.
Make sure that the code only works with STEP and DIR commands. Here you will find a simple example to get first moves. You can use different pins, but I adjusted my example of the wiring with @Robin2's simple stepper program.
But, before the motor moves, it needs to be enabled by setting the "ERC" pin (=ENA-) of the driver to "LOW".
There are 2 alternatives for that:
a) you declare pinEnable as pin#10 in the declaration section, set it up as OUTPUT in setup() and HIGH as initial value first; after that set it LOW whenever you want that the motor moves when it receives STEP pulses.
b) To ease your test you could connect the ERC instead to pin 10 of the Due (in my example above) to GND.
This will permanently enable the driver and the driver will continuously drive the max. current (which you hopefully have limited at the VERY FIRST STEP of your experiment to the max. current of your stepper motor !!!) through the motor - regardless if it stands still (no pulses coming in to STEP-) or if it is turning.
The timing diagram on the right of your image shows the interdependencies of the signals. It says that there are some (internal) delays which happen when you change either ENABLE status or DIR status.
If you immediately send step pulses after you have sent either ENABLE or new DIR command, you will lose the first steps until the commands take effect.
In the operational environment this will mean that you will have to wait some us until the driver has realized/processed the change - but we are talking about us-range - might be that due to the slowness of the Arduino Due you might not have to consider this in your code. If it becomes critical, just insert 1ms delay after each change of either direction or enabling. Nobody will notice it and your stepper won't miss any steps.
Good luck.
EDIT:
@Robin2 was quicker (I had to finish the novel first ..)
It would be nice if you could provide an exact link to your driver or it's data sheet so we could look it up for either confirming the "external transistor non-issue" or not.