Make sure that your frame doesn't impede your pulling units in and out. Since this appears to be interior/conditioned space a louvered door would be a better option. You don't want to trap heat and humidity in a closet. Is there a drain in the floor? I've had a washer in the kitchen and I would never have one there again.

What is the depth of the washer alcove?

For full-size stackable washer and dryers, you'll typically need a space that's 80" high x 28" wide x 31-34" deep.

You need an alcove that is 31-34" deep plus another 5" or so for the door jambs. It doesn't appear that you have enough space as is. The walls will most like need to be extended farther out into the room.

ETA--That, or the compact line of washers & dryers.

"I ask because we don’t have our stackable washer and dryer yet"

1. I wouldn't do anything until those appliances were installed to make sure that you don't mistakenly block their installation or access. Appliances are getting larger and larger.
2. If you install a door, will you be ok with the amount of corridor space taken up by the door when it is open and any inconveniences it causes when using the current and future appliances?
3. After their installation, you may find that the stacked appliances look satisfactory and you no longer want a door.

for where you have it, I would make this just like a utility close w/a hinged door that swings out to the right. That's the only thing that's going to look decent here.

if you could do a wider space, you could do this