If the drain top or screen is the screw in type, then a good bead of silicone will seal the screen to the rubber gasket, if the screen is plastic, it is meant to be glued to the ABS pipe and the rubber seal is supposed to compress to keep the water in the drain. There has to be a way to mechanically keep the drain in place, and you cannot rely on silicone to do that. It may have been put in incorrectly with plumbers caulk, but that is not as good as what was mentioned. You may want to take the screen part and the seal to the plumbing shop and find out if a part is missing and someone fudged the job. It should be an easy fix once you know which system you have and put it together correctly.
If bbbs53's answer doesn't work, it's a possibility that you shower pan is cracked or broken. This can happen even with very new showers and go undetected while you think that the trouble is the drain or tile damage or any number of other things.
Put a thick clear coat of silicone around the pipe then put the cover plate back on, then wipe off anything that squeezes out
Dry it then get you some rubber sealer and go around the drain