We highly recommend you do not attempt to clean opals, emeralds, or other very soft precious or semi-precious stones at home. These stones are very sensitive and if you're not careful they're very easy to damage permanently.
Take Your Opals & Emeralds To Your Jeweler
Jewelers are experience with the characteristics of different stones and how they behave. They also have the insurance to cover your jewelry if something happens to it while cleaning. This is the safest way to get your opal, emerald, or other soft precious stone jewelry clean.
Most jewelers will clean your item for free or at least a very nominal price. This is also a very good time to have the jeweler check the prongs on your jewelry for security and tightness.
If You Really Must...
Clean your soft stones at home, and again we do not recommend it, only use tepid water, a soft toothbrush, and non-harsh dish soap like Dawn. Be ridiculously gentle when cleaning. Don't use any chemicals, cleaners, or vinegars. Don't use steam. Don't use an ultrasonic cleaner.Don't blow dry the piece when done, just allow it to air dry.