To supplement what they wrote, gravity is your friend, here. Although the moisture under the sheet will be there, it will "want" to go down with gravity, not up against gravity, unless there is some sort of forceful pressure pushing it up from below or it tries to evaporate into drier air. The plastic sheet creates a barrier that serparates the moisture in the ground from the air in the basement so that it does not evaporate. In other words, the sheet will merely let the water "know" that it can't evaporate there in the crawl space and should just keep going down with gravity to the water table.
Using a dehumidifier will help eliminate the musty smell, too. It will dry the air faster than the moisture evaporates upwards. Again, only a minimal amount of water should normally get there because gravity is pulling it down, not up.
Cleaning the crawl space and other spaces with bleach every once in a while will also kill any mold causing the musty smell. IF it is a disgusting crawl space, all the better to clean it, even if only to dump bleach on it and let it soak/penetrate into the foundation and kill 100 years of potentially built-up mold.
And, yes, you should patch any finger-wide holes in your flooring. Chances are that foot traffic will splinter the floorboards and make the hole bigger with time. Moreover, it is always best to seal the crawl space off from the living space.