How did the prepositions up and down both come to be associated with writing?

Most English prepositions and phrasal verb particles have some intrinsic physical meaning, usually from the perspective of a human body. For instance,

• We refer to something or someone being located
  at a point, on a surface, or in a container
• We refer to something (the trajector) moving
  from one location or actor (the source)
  to another location or patient (the goal).

All of these have many other, often idiomatic, metaphoric, and non-physical meanings, too.
And there's loads more prepositions and phrasal verb particles left.
But I'm not going to deal with them here; life is short.

This is only about up and down.

Up and down are directions, but they're directions from the viewpoint of a human. Humans have evolved and adapted to a 1-G environment; we have built-in senses to detect up and down; it's important to humans.

For example, imagine a standing human:

• Up means the direction from the feet to the head, and continuing beyond.
  Down is the opposite direction.

It is crucial to our species that we can move freely in the first two dimensions -- on a surface -- but we can't move freely in the third, up/down, dimension. Birds and fish have true three-dimensional motion; humans have, at best, 2½-dimensional motion.

But we can look up, even if we can't always go up; and we can tell when something's there. Whether it is or not. That's the source for most of the metaphoric senses of up and down,
which are frequent visitors to phrasal verbs.

These senses often converge on the same situation, as in

1. He drank it down ~ He drank it up
2. The house burned down ~ The house burned up
3. I'll write that down ~ I'll write that up

In each of these, the meaning is not quite the same.

The down cases refer to Physical down --

• the beer is now down in the stomach;
• the house was high, now there's only ground left;
• the writing has been physically put down on the paper
  (this is also coherent with DOWN is CONCRETE;
  writing is very concrete, compared to talk)

But the up cases refer to Completive up. That's the up that means 'to completion; all the way'
You get it in phrasal verbs like
tie (it) up, pack (it) up, finish (it) up, screw (it) up, which mean
to do whatever the verb phrase is as much, as thoroughly, and as completely as possible.

In that context, it's easy to see that beer that is drunk totally has moved downwards,
and that a house that is completely burned is burned to the ground,
and that something that has been written completely is written down on some medium.

There are situations where they fit in different configurations, and -- depending on context and idiom -- you can sometimes see the cracks where they don't quite fit.

As one example, to write something up means to produce a finished version, while to write something down merely means to make a note.

For another, only constructions that are rooted on the ground -- like houses -- can burn down or burn to the ground; if an airliner was completely destroyed by a fire while on the ground you could say that it burned up, but not that it burned down or burned to the ground.

For still another, drink up is coherent with the physical upward movement of the drink container to the lips, while drink down is coherent with the reduced vertical level of the liquid in the container after drinking.

As to when this happened, there are dates in dictionaries; they represent first appearance in surviving text, which may or may not represent actual speech. But the meanings are primal enough to occur to anybody, and to be easily passed along to the next generation where they hadn't already been figured out. Like a lot of things, they keep getting rediscovered and reinforced.

So it isn't that up and down are associated with writing as such, it's just that they form pretty typical phrasal verbs with write, just like they do with thousands of other verbs.