If water is collecting under the plastic and not being readily absorbed by the ground, you have a number of things to look at.
During the next heavy rain storm, get an umbrella and go out to see if your gutters and downspouts are properly working. If they're over-flowing, the ground may be too saturated to hold all the water, thus, water in the crawlspace (even under the plastic). If the gutters and downspouts are in proper working order, is the water that comes out the downspouts pooling near the foundation? If so, this too can contribute to the aforementioned. The water needs to get as far away from the house as possible. Also, check for negative slope. If any of the landscaping around the house PROMOTES water to drain TOWARDS the house, this can contribute to the aforementioned as well. And lastly, you mentioned that you have vents. What size is the crawlspace (in sq. ft.) and what size are your vents and how many (3 questions).
With properly working gutters and downspouts, with water not inadvertently being diverted TOWARDS the foundation, with the proper venting for a crawlspace, you can consider a sump pump. This will take some doing and isn't exactly inexpensive, especially if the dirt needs to be graded in the crawlspace and such. (Get 2 or 3 bids to be sure of this.) Oh, as an afterthought, if the landscaping, gutters, downspouts, vents, etc., are in proper order, maybe there's vegetation obstructing airflow into the vents. IMO, 3-6' minimum on either side of the vent is what you want. If the air around your house has a tendency to 'run along' your foundation, then bushes and stuff can prevent it from getting to the vents. If the air has a tendency to flow perpendicular to the vent, then the 3' minimum should be OK.
My best to ya and hope this helps.
PS: At worst, if you get bids, consider the expense to that of your health and the health of the home.