I have a fairly large deck about 5 feet off the ground.
Right now weeds grow under about half of it, and the other half is dead dirt.
From looking around, I think I have a few primary options.
Mulch it. Probably with plastic underneath. Still have to weed it occasionally, as weeds will eventually grow in the mulch or in dirt that gets blown into it, and I'll have to re-mulch it every few years cause mulch breaks down. I've never once had luck with landscaping fabric, but I worry about plastic and water runoff. This is probably fairly inexpensive, though I am quickly getting too old to enjoy mulch, shovels, wheelbarrows, and rakes. The area would probably involve 2-4 yards of mulch, maybe more. I don't mind grass meeting the mulch as I can't get my mower there anyways and I don't mind using my trimmer to keep the border tidy.
Put down stones. Probably with plastic. Same as above really. Maybe less maintenance, but much more labor to install. More expensive if I hired pros to do it, because stone is so much heavier?
Try to get some low-light plants to grow in controlled conditions, and mulch or stone the rest.
I have mixed feelings about landscape cloth or plastic but will say, that like many other things, it depends on what you buy and what you pay for it. First red flag that popped for me was the possibility of soil and seeds blowing in on top of it.
You might look into rubber mulches. Some are quite natural looking. They organics in them will break down eventually but no so fast as wood mulch. They come in a wide varieties of colors and textures. A few argue they will leach hydrocarbons into the soil. We put a lot more dangerous stuff in higher concentrations---like heavy metals---just through granular fertilizer application and "anything goes" under inert ingredient labeling that does not require specifying beyond that.
It would be helpful in this instance if you could snap photos of the specific problem area and an overall shot of how the deck and the space under it fits into your existing landscape.