Your situation is not uncommon.
Here's a study on crawl space encapsulation in the Southeast USA: http://aceee.org/files/proceedings/2004/data/papers/SS04_Panel1_Paper07.pdf (Moisture Solution Becomes Efficiency Bonanza in Southeastern United States Bruce Davis and Cyrus Dastur, Advanced Energy #DE-FC26-00NT40995 ). And see also http://diy.stackexchange.com/questions/3666/what-are-the-pros-and-cons-of-a-vented-vs-unvented-crawl-space?rq=1What are the pros and cons of a vented vs unvented crawl space?
Don't do the walls with anything impermeable: you'll trap moisture in the space which is worse than doing nothing. Do consider doing the floor. Using bubble wrap below the floor can reduce tearing of the plastic. Note that some people worry about plastic leading to mold growth below the plastic: thus an effort to completely seal the ground.
You should get a humidity meter (they are not expensive) and monitor before and after. Report back with measured humidity levels now and it might help refine the answer.