The french door would be nice. We put in a frosted full panel between the kitchen and the laundry room. I can't explain but there is something about the glass that makes it seem like the space is bigger.

You could either go clear, or like you said the rain glass or even frosted, with a light behind it, it looks really cool.

I know Lowe's has the various glass options. We bought ours at a discount place, they are ones that might have had the hole cut on the wrong side for that job, etc. They also sell ones that are new which are still 1/2 the price of Lowes'. Mine was about $100 total, including the holes and hinge thingys being cut. Check with smaller lumber places, the place that the builders go. You might find what you are looking for.
My door:

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We have a french door with a white linen curtain going into our bedroom - it obscures the view but helps it blend with the other doors. It is a nice airy look - my camera is on the fritz so I can't take a photo (sorry!) I think a white rice paper shade could also be nice or use fabric that would tie into your kitchen colors. It would certainly dress up the entrance to the space.

Tightly shirred sheer curtains are a classic french door treatment. (Rod top and bottom with fabric stretched tight)

My neighbor has a french door leading to her basement and it looks great! When we finish our basement I plan on having the same.

I changed out my regular six panel door to the basement for smaller french doors for two reasons - first, for safety: if someone was coming up from the basement and the door was closed they couldn't see if anyone was was on the other side which had resulted in a few thumps and second, I wanted the kitchen light to help keep the basement lighter and lessen that "basement darkness". I am still surprised how much brighter it looks while downstairs just from allowing that light to flow down.

I have travertine in my kitchen that moves to carpet on the stairs and basement - not a big deal at all.

Hope this helps!!
Beth

Check local code before you make any changes. Our community has created new rules that preclude a door swinging into the stair area. We don't want it swinging into the room because that puts it intrusively into a walkpath to other parts of the house.
So... we've put in a new pocket door assembly in order to make things legal and less hazardous--even if it were legal to swing a door inward into the stairwell, the doorknob would still be precariously out of reach because it's a big door. We haven't bought the new door yet and I don't know what it's going to feel like to pull the door left and right, but we did put in high quality hardware so it should move easily. One thing that I hope will be doable is to leave the door half-shut, to partially intercept the intruding sound of our geothermal unit (which sits at the bottom of the stairs) and still allow the dog his access to basement.

I am considering all kinds of door styles and am very drawn to ones with windows that would allow light into the dark basement even when door is closed. DH is big on a different door. Not sure who will prevail.

As for the question of flooring differences, I suggest that the top step have the same treatment as the floor that leads out onto it so that there is no hump of carpet on that top place. A wood floor is a good choice here. After that, the steps can be carpeted. If your community has very stringent rules about consistent stair height, you may want to reconsider the carpeted steps, though, unless that carpet is very low. I live in a community with a proactive (intrusive?) set of home regulations and they have been a surprise at some points--ouch! Don't let yourself get caught unawares.

We bought divided light french doors for our pantry and then frosted the glass ourselves. We also put in two recessed lights to back light them.