Learn how to install a blade into a jeweler's saw with this two-part video tutorial. Make sure your jewelers saw blade is facing outward and the teeth are pointing down. Run a finger over the blade before inserting it to make sure it is in the right position. Once it is in correctly, tighten the thumb screws at the top and bottom first.
Once the blade is secured tightly in the top and bottom of the frame with the thumb screws, push the back of the frame with your thumb. At the same time, tighten the back thumb screw. You should hear a high-pitched "ping" when you pluck the saw blade. There shouldn't be any vibrations. A loose saw blade is much more likely to break. You may need to retighten the saw blade frequently as you saw.
Again, remember to have the saw blade pointing outward are the teeth down. You'll be able to tell if it is in correctly if you run your finger over the blade. If it is smooth, you have it in wrong.