If the crawlspace has air-communication paths between it and the basement, the basement is essentially at outdoor humidity levels due to those exterior vents in the crawl. Crawlspace ventilation to the outdoors was conceived to be a way of letting ground moisture escape and put into building codes on this theory. While that theory may hold true in cooler drier climates, in places hot and humid as FL venting crawlspaces to the outdoors INCREASES rather than decreases the average crawlspace humidity. Read
this article for details on converting itl into a sealed-conditioned crawlspace (and why.) You may be required by code to keep the vents ,making them operable, opening only when you need to for some reason (say, for drying after a bulk-water flooding event.) Insulating the walls with an inch or two of XPS rigid foam (pink/blue, not beadboard or any poly/foil faced stuff) rather than merely vapor-sealing them with poly will earth-couple the building, decreasing both the sensible-AC & heating loads. Either leave a 3" gap in the insulation between the top of the foam and the foundation sill as a termite-inspection strip (those suckers will tunnel through foam sometimes), or nail 6" of copper flashing to the foundation sill extending down between the foam and the foundation to deny them an unseen path to wood.
The plastic tarp on the floor of the crawl is a good start. If it's long enough at the ends, sealing it to the foundation with duct-mastic before installing any wall insulation or vapor retarder will increase it's effectiveness once the vents are sealed.
The vinyl flooring in the finished basement is an effective capillary break & vapor retarder for ground moisture on the floor. On the sub-grade finished walls behind the drywall let's hope they at least applied some masonry sealer to lower the capillary draw, but do NOT put any vapor retarders such as vapor retardent paints, vinyl or foil wallpapers on the drywall or you'll create mold condtions in the studwall.
Where the exterior walls are exposed and completely above grade, acrylic or silane/silcone based masonry sealers both inside & out will reduce the capillary draw from any rain or dew -wetting yet still allow the masonry to dry from vapor permeation. When sun hits wet masonry the vapor drives skyrocket. Insulating the interior with semi-permeable or semi-impermeable foam (1-2" of XPS), or impermeable foil or poly faced EPS beadboard or iso reduces the amount of vapor permeation through the wall into the basement. (Again, don't use the impermeable stuff below grade, or you'll end up with higher moisture content in the masonry, which could rot the foundation sill and/or cause the above grade portion to have efflorescence & spalling issues.) To meet fire code, foam insulation needs a thermal barrier against ignition- half- inch drywall works.
Gluing the foam board to the masonry with purpose-made foam-board adhesive in a ~18" pattern of walnut sized blobs, then through-screwing thin furring (even ripped down 3/8" plywood works, or standard 1x furring is fine if the wall thicknes is OK with you) into the wall to hold it in place gives you something to mount the drywall to. (Ignition barrier may be required in the crawl as well.) Devising a spacer method to keep maintain a 1/4" gap between the masonry & foam will allow it to operate as a cavity-wall, giving those high vapor-drive conditions a place to vent the moisture to other than the foam, and any liquid water that finds it's way in then falls to the floor. This is more important if you use vapor-impermeable rather than semi-permeable foam, and more important on sides where the sun would actually hit a rain or dew wetted exterior wall. It's less critical elsewhere- the glue blobs usually give you about a 1/8" cavity if you don't press too hard, and wait for it to set up before applying the furring for the wallboard.
A window fan can be used to pressurize/depressurize the basement to find & fix all of the air leaks. Stack effects in even 2-story houses are significant, which will draws air out of the attic depressurizing the house, drawing air into the basement. When hot the humid outdoor air drawn in cools in the basement & crawl, it's relative humidity goes up, which is likely to be more than half your basement humidity problem. Blocking the flow into the basement (and out of the ceiling on the upper floors) breaks up the stack effect slowing the flow, reducing both latent & sensible AC loads.
Air sealing the ceiling to the attic is step #2, but with ducts in the attic it's often both easier & better to foam-insulate the underside of roof deck with 4-6" of open cell foam & close off the attic ventilation (especially if this is a circa 1980 house with the typical 27 recessed lighting cans that would need to be boxed over to air seal it.) A ceiled insulated attic puts the ducts completely inside the pressure boundary of the house, reducing air-handler-driven air infiltration, and rendering duct leaks relatively neutral. In a ventilated attic sealing & insulating the attic ducts is far more important. Radiant barrier between the ducts in addition to duct insulation is also worthwhile. (If you ever go the sealed attic route, don't let some foam installer tell you that you need to remove the attic floor insulation, or that "(x)-inches of foam is all you need". R30 of foam is expensive, and any fiberglass or cellulose/whatever on the attic floor will continue to provide signficant benefit, and causes no problems in cooling dominated climates. In cold climates you'd have to play the R-ratio game to limit wintertime condensation potential, but not in FL.)
[Edited to add] Another easy-reading discussion of vented vs. unvented crawlspaces lives
here. Even if you don't have mold problems currently in your crawl, it's undoubtedly adding to the latent-load on the AC and keeping the basement at higher humidity in your climate zone.