There's different types of "clear" enclosures. What type do you have?
The real trick with caring for these is to not let it get so bad to start with and stay on top of keeping them clean and protected. And use the right products for the right enclosure material. For example, don't use Windex and paper towels. Most clear vinyl (or acrylic/whatever) manufacturers have specific recommendations for care and cleaning.
Since it's at the point that it is now, you can't really lose anything by getting aggressive. Personally, I've never had to go to this step but I remember reading that a metal polish (Flitz, for example) can work on various vinyl products (Eisenglass, Strataglass, etc).
But again - first you need to know what you have.