To what extent is the openness/closedness of positions determined by white? By black? Say I am playing as white and I strongly prefer closed positions while my opponent strongly prefers open positions. To what extent do I have control over this? Will the game more likely end with a closed position or an open position? What are my strategies if I want to keep the position closed?

If I am playing black, does this change the answer to the questions above?
In general White has more freedom of choice due to going first. He can play a move that is slightly suboptimal and still be around equal. But if Black does the same thing, he could quickly find himself in a bad position.

However, it's obviously not always necessary to play dubious moves in order to get the kind of position you want. In general you should play openings that lead to closed/open positions. In the former case 1.d4 makes more sense, and 1.e4 for the latter. As Black, you could meet 1.d4 with something like the Grunfeld if you like open positions, or with the KID if you prefer closed positions. For 1.e4, the French/Caro lead to more closed positions, while the Sicilian has a higher chance of leading to an open position. Of course, these are just examples.