How do you feel about no window treatments at all? Your yard appears to be fairly private. If privacy is not an issue, you may really enjoy the extra light. I went in that direction with a kitchen remodel several year's ago and do not regret it. However, the remodel did include replacing the sliding doors with new doors containing low E glass which makes a huge difference.

If the doors are well-used, you probably don't want to go with drapery. Hunter Douglas makes 'soft' vertical blinds, in many shades and colors, cloth-covered vinyl (I think) in their Cadence collection. They also have vertical wood blinds in their Crosswinds collection, if you want that look. My DH used to install H/D products; they are a far cry from the vinyl slats that you get at a home improvement store!

I always like drapes because you can switch them out in a couple of years if you get tired of it and they're usually not too expensive. I can't tell what is on the left side of the door, but I think if you decluttered that area and put some light colored drapes up it would look fantastic.

you can get grommet draperies that would slide one way so you can still go in and out of door, looks like you have a window to the left so you could put a matching drapery pair there

Wow! Thanks for all these ideas! I think I have to stick to some kind of window covering since come winter time, most of the trees will be bare and we won't have quite the privacy we have right now. Thank you!

I like the idea of a fabric wrapped vertical blind for this patio door. They will stick with the present decor in your dining area. Open, these window treatments look just like a sheer and let a lot of light in, but closed, the pvc (fabric wrapped) fall almost flat and tight together, which will provide you with the privacy you seek. :)

I like wreaths idea. We had sliding doors in our dining room. I replaced them with french doors which look lovely opened during warm weather. Looks like you have the space on both sides of the walls for the doors to open. Additionally, I have blinds inside the glass which serves as privacy and you don't need to clean them! The room looks bigger, but you can always add drapes later if you wish.

I like the "no drapes" idea, but you asked for drapes. Forget symmetry. Install a curtain rod all the way to the left, so that curtains when open, are against the left wall, and you get maximum use of light and the door. There will be no curtains on the right. Depending on the fabric, you'll need about 3 feet of wall space so as not to cramp the curtains. Remember to choose rod/rings that make sliding easy. Rod pocket curtains wouldn't slide.

Additional comment: get rod as close to ceiling as possible and make the curtains floor length, and choose slim scale rod and sheerer fabric. This meets two goals: not crowding the room, and make the doorway seem bigger.

Sliding Panel Systems are very popular for sliding glass doors and you can coordinate with the same fabric in roller shades if you have other windows nearby:
Another way to soften the look would be to get custom-made ripplefold sheers

I have the Sliding panel system in family room-really like them. I did make sure they were wide enough to completely stack off the glass, so no glass was covered. They are in wood look and on adjacent window had same wood roman shades done.

Bonrud, could you please post pictures?

The real wood (and faux wood) linds and shades come in a wide range of finishes from very light to very dark.