The use of down implies a physical activity, while omitting it implies a result or activity that's somewhat removed from the physical (deleting something).
✔ I am going to wipe the computer's memory.
✘ I am going to wipe down the computer's memory.
In this example, it doesn't make any sense to use wipe down with computer memory, because that's not something that can be physically touched.
✘ I am going to wipe the car.
✔ I am going to wipe down the car.
Here, I would have no idea what wiping a car is—because you can't delete cars. On the other hand, wiping down a car generally means applying a cloth to remove dirt or water.
(Note that we often omit prepositions in speech. if somebody actually said wipe the car, I would understand from context that they meant wipe down the car. As such, the expression without down is not actually wrong in regular discourse—but it wouldn't be quite as natural, and I'm trying to show the literal difference between the two.)
To be more specific, compare the following situations.
Sitting at a computer keyboard and typing commands:
✔ I am going to wipe the motherboard.
✘ I am going to wipe down the motherboard.
Holding a motherboard and cloth in your hand:
✘ I am going to wipe the motherboard.
✔ I am going to wipe down the motherboard.
The object in the question appears to be a blackboard, on which there is writing. This is a particular instance where the two different senses of wipe can be applied to the same thing.
It can accept wipe in the sense of removing information. It can also accept wipe down in the sense of cleaning.
✔ I am going to wipe the board, so write down what I have written.
→ I am going to remove (delete) the information from the blackboard.
✔ I am going to wipe down the board, so write down what I have written.
→ I am going to clean the chalk off of the whiteboard.
So, both are fine. Which you use is based on the specific sense you are trying to convey. Are you talking about the information represented by the blackboard or are you talking about the physical chalk marks? (Note that the final result is mostly the same, with the one sense impacting the other.)