Thanks for the added info, Gary; it sheds new light on possible solutions.
When the soil outside is moist, but the crawl is extremely dry, it's often a recipe for foundation problems; this is a situation where putting plastic down will help the soil retain a better balanced level of moisture. In an overly damp crawl, you want to first get the excessive level dealt with by means of added ventilation, grade corrections, sumps, etc. Plastic under these conditions will only hamper the effort. Once the water situation has been corrected and stabilized, plastic is fine.
Ideally, you want the moisture level in the crawlspace soil and outside soil to be reasonably "in sync" with each other so that moist, expanding soil outside trying to apply pressure to the foundation is counteracted or nullified by moist soil inside, pressing back.
As to the concrete, personally I don't think it's the right solution, and would be pretty pricey, too; you would have to have a concrete pump for placement, and placement in a crawl is definitely not be a picnic. You would also probably have to do any smoothing, leveling or finishing as you went, working your way out, since you could not work a bull float very reasonably in a crawlspace. Down the road, the concrete might also be an interference with plumbing repairs, etc. where you might have to access pipes below the slab.
Overall, I think it's initially an avoidable and unnecessary task and expense. I would first take a different, less drastic, and certainly less costly approach; only after finding this unsuitable would I go to more extreme measures.
Gary, your first line of defense here will be to further examine your exterior grade and guttering for any possible improved watershed. If you have room to do so, additional soil can be added outside to improve drainage. You'll want to keep your siding and crawl vents at least 6" above the soil line at the house, and maintain a comfortable but definite slope (1" per foot is good) away from the house for 10' to 12', which can then become more gradual or almost level. Driveways and sidewalks may control how much can be done in some areas; handle these with subtle swales and berms in your grading to direct the flow of water as needed.
Splashblocks are often a source of trouble and downspout extensions are needed, but with an improved grade the splashblock may be all that's necessary. Make sure they are in place and shedding water correctly by watching the downspouts while running a garden hose in the gutter above. It won't match a heavy rainfall, but it can point out possible need for changes.
Now to the crawl itself; if you have adequate ventilation and at least SOME grade away from the house, I think installing a sump will resolve your problem. If you also are able to make exterior improvements in drainage, this sump may only be needed for the initial adjustment period.
Depending on how involved you wish to go with it, your sump could consist of a standard sump well (the collection barrel, such as you would use in a basement), fed by a run of sleeved, corrugated plastic drain tile trenched into the crawl near the perimeter, OR, as little as a rigid plastic bucket or trash can, slotted with a skillsaw and wrapped with shade cloth or mesh weed barrier, buried in an oversized hole, backfilling the sides with coarse sand.
I would choose a fairly central location, away from vents if freezing were a concern, but within reasonable location for a straight shot out with the discharge pipe, and relatively easy access for any maintenance or inspection. You'll want to use a check valve and rigid pvc pipe for the discharge; that coiled black hose will sag and remain full of water, possibly freezing. Make sure to slope your discharge pipe so that the horizontal run leading to the exit drains itself, and outside use a splashblock or extension to ensure the sump water is leaving the area.
Once your excessive moisture level has been relieved, you may want to insulate your joist spaces (with any vapor barrier placed towards the living space), and plastic drape the soil after it has proven to remain dry.
Sorry this almost became a small novel, but hope it will help you in your decision. Good luck, Gary, and let us know how things turn out.