If this is a crawlspace that is never entered (or maybe only VERY rarely like for fumigation or plumbing repair say), then a heavy-weight plastic crawlspace vapor barrier sheeting (commonly available in 12, 20, 30 mil in my area) can be put down. It needs to be sitting on material that will not puncture it during light maintenance as described above - meaning either a fine dirt with all the rocks and pebbles pulled out, or a rounded (not crushed) sand layer 1-2 inches thick. 12 mil will NOT survive being crawled on unless it is covered with protection like styrofoam board or sand - 20 mil is OK but will tear if jabbed with a tool or belt buckle by someone crawling, so I recommend 30 mil. of course, if you are looking for insulation too, then rigid foam board works well for that.
The plastic liner has to be waterproofed at the foundation wall, typically with an asphaltic mastic or other adhesive recommended by manufacturer (type depends on type of plastic), and adhered to the mastic with wood battens screwed or expansion bolted into the foundation wall. It is common to put granular pesticide along the base of the foundation wall just before sealing the edges of the plastic, to stop insect nesting or tunneling under the plastic in case insects get in there. BE SURE TO PLACE A WARNING SIGN at the entrance to the sealed area about the pesticide under the plastic, as it can stay active for a long time due to the protection from weathering.
If you walk around in the basement or use it for a storage room or workshop or such, then for very light use like occasional storage you can use a heavy-duty (typically 60-120 mil) pond liner like Hypalon - the sort of liner used under landfills and containment ponds. For substantial or regular use you would then have to cover the liner with a protective material, in which case you would be better off and almost always cheaper to just put down a vapor barrier rated for under concrete (minimum 10 mil per code, normally 12-15 mil polyethylene) covering the entire area, then sealed concrete over that. If the moisture issue is due to just migration of moisture through the soil but not free water at the concrete level, that might take care of it.
If your moisture issue is due to high groundwater (due to being near river level) then if you do the above you might still get water coming up around the edges of the slab. In that case, installing a french drain collector about a foot below the vapor barrier around the perimeter draining to a sump pump, and also either free draining gravel or drain lines about every 6-10 feet across the width, might be needed to lower the water table enough to avoid moisture migration into and through the slab.
The decision on underslab drainage can take a bit of thinking in wet conditions - if you are close to the river AND in free-draining soils, then you might be introducing a river into your basement with the underdrain. In that sort of case you need to either injection grout under the foundation to provide a cutoff to most of the flow, or in "swimming hole" conditions, bond the slab to the house foundation to prevent leakage. Before doing that you need an engineer to evaluate the situation first, to be sure you will not float your house out of the ground during high water table events, and the slab needs to be thicker and structurally reinforced to handle the pressure under it.
If you do not have free water flowing into the basement now (or during high river levels) then the underdrain system should do the job - dug down a foot to keep the water level BELOW the vapor barrier and slab level, then free-draining gravel under the vapor barrier, and concrete over with mastic waterseal at the foundation joint, then double-coat concrete sealer on the slab. For very light use in the area, just the drain pipes with sump pump and vapor barrier over the existing ground, then a sand or sandy clay layer over that to protect it from light walking might be adequate. Whether the drains will work in your soil can be tested by digging a test drain a few feet long about a foot deep - if free water flows into it, they will generally work. If the soil is so clayey that no free water forms in the ditch but the soil is still damp or wet, then drain pipes will likely not help much, but a sand bedding bedding with vapor barrier should control the moisture issue. To be on the safe side, the existing grade should be graded to drain to a corner where a sump pump could later be put if needed, in case free water starts accumulating under the vapor barrier.
One other possible solution, which is generally more expensive, IF you have enough topographic drop on your land so a french drain at the foundation base level can be drained to free surface away from the house, is to excavate around your foundation (at least on the wet sides) and install bitumastic sealant on the foundation face to keep the wall dry, and a french drain at the bottom of the footer level, draining to free ground surface. This will intercept water before it gets to your basement. However, in areas with high water table, the french drain either needs to be several feet BELOW foundation grade, or you need the drains under the slab also, because the groundwater will still come up under the slab if its natural level is significantly above footer level.
So, it all comes down to how much you want to control the water and if you need to be able to use the basement or not.
You can see the recommendations in several other similar cases in the links right below these answers.