The basement floods every few years (a pump gets confused and lets water in; can't fix that reliably). Carpet gets soaked. I think this is why my wife starts sneezing when she's been in the basement for an hour or so.
I think the carpet has to go.
We're looking online for ways to make concrete floors look beautiful (or at least not so industrial). What's a good cheap route? We'll use rugs, I think, to keep our feet from freezing; but it would be nice if the floor itself looked OK.
You could stain it, or acid etch it, there are also epoxies you could apply. Those options would be the least likely to result in further water damage. A good alternative is industrial tiles, though water damage may still occur with that.
We painted our concrete basement floor; the easy way out. Other options are as Shane said, but that does take more work.
We did try some carpet tiles, easy to take up and hose off when we do get some water in the basement.
Then the cats pissed on them and we had to throw them out; another problem altogether.
Definitely want to stain, paint, or etch it. No covering -- it will flood.
The stains work well.
Look up sponge painting and there are some 'graining' tools you can buy at any of the chain hardware stores. . . with some practice can get you a faux wood grain look. So it would 'look' like a hardwood floor.
I love that idea.
Of course if you know someone that paints theater sets. . . you could bribe them to do it freehand.
My brother-in-law. He doesn't like me, but he likes her...
It's wise to keep in-laws out of home remodeling projects.
First I would recommend replacing the pump and float (if they are separate) and adding a battery powered backup pump. Also have the drain tile around the house checked for damage and blockages and replaced if necessary. It can be expensive but well worth it if it keeps water out of the house.
For the concrete I would acid wash to remove contaminants, stain, and seal. You can use painting techniques such as blocking or marbeling to make the floor look interesting.