Most likely you have a wood frame against the concrete into which the doors have been installed and onto which the drywall has been attached.

Drill a few small pilot holes and/or use a stud finder to verify this.

If you locate a wooden structure you will be able to install trim molding or anchor window coverings through the wall and into the wood. Hope this helps.

It doesn't look to me like you have any other choice but attach rods to the wall outside the recess since you don't want to put holes in the doors and you don't have any clearance above the doors under the recess. I actually think that would look very nice.
They do make miniblinds that are encassed inside glass that can be added to existing doors though. Yes you'd be putting holes in the doors, but it would be for a permanant instulation. We have built in minis on a couple of our doors and I wish we would have had it done on them all. Just never thought of it at the time.

You know, until you mentioned it, I hadn't thought about the clearance of the door opening, I'm glad you pointed that out!! I was focusing more on the how to part and the look of the doors. At least I can quit looking at that option now. Some trim around the door will look nice, too. I'll add that to my to-do-list. I think I'll also look into the built-in mini blinds. I wouldn't mind putting in holes for something permanent like that. And the more I think about it, I'm liking the idea of curtains on the outside of the recess (on the wall itself). It's usually good for me to think about things for a while. My dh accuses me of doing things too fast and then changing my mind in a few years.

Because the doors are so long and you want anything that covers the doors to be able to clear the doors to allow the doors to open, I'd mount whatever you decide on to the ceiling of the recess. You could use vertical blinds that open from both sides. They would be able to stack in the small space next to the doors. If you do decide on miniblinds or honeycomb blinds, you're best off with at least two, to minimize the weight. If you don't mind covering some of the fixed panels, you could use draw drapes.

They make enclosed mini-blind kits that attach to the exisiting window frame on your doors. I had them on my doors in the past and they were great. I found them at Home Depot (see link, below). If I remember correctly, they didn't require any drilling, either. When I eventually replaced my patio doors, I bought them already built-in.

Here is a link that might be useful: Patio door blinds

Panel track can fit in application where there is not much clearance.

Here is a link that might be useful: Panel Track

I saw something like Panel Track on the Ikea web site. It was something that started with a K. I don't have an Ikea where I live, but there is one near my brother in Atlanta. I need to get some more information on the clearance/depth of the track. That window/door in the picture looks exactly like mine! Thanks, dilly.

I see that Lowes also has the patio door blinds like HD (thanks, charlieinnj), so I might check them out this weekend. That is a bit more than I wanted to spend, but long term might be the best option.

I haven't quizzed dh on his preference yet. Now that I have some options, I'll see what he thinks. With the pool table and wii we're getting the kids for Christmas, the basement may turn into the 'man cave'.

I thought of tracks too since you can get them to hang curtains too, but it doesn't look as though you'd have enough clearance.
As for the built in minis I mentioned I don't know what they are called, but I've seen them at HoDe and Lowes so you can ask there.

I thought I would post an update of what I've done so far (so much more to do!!). I did this before Christmas because I needed to cover the windows so guest could sleep in the basement. The doors are fiberglass, so this is a gel stain on them. A friend has suggested that I put some stained luan on the recessed part and frame out the doors (molding) on the outside wall. I really like the idea, but I'm waiting for some warmer weather, so this is what I have so far...

{{!gwi}}

Wow! That looks great. It really warms up the room. You just used gel stain on the door? Did you just paint it on and let it dry? And, are those jingle bells on the door for you furbaby? We have a similar set on our door, and that's what it's there for.
Dee

Yes, that's gel stain on the doors. I'm not sure I like the stuff, but you pretty much follow the directions on the can. Paint on and rub off the excess. If you want a darker color, you can't just leave it on, you have to do multiple coats.

LOL, on the bells! Yes, they're bells, but just for decoration. I guess they may fit the horses (definitely too big for the dogs), but we've never used them for that. I thought I'd leave them there so I'd know if the kids went out that door (cheap alarm?)

Thanks for the info! Our dogs don't wear the bells; they ring them to go outside. Well, one of them anyway. Our rescue Aussie thinks it's beneath him to ring bells. LOL.
Dee