In theory, since the inside of the cage and the outside are completely isolated at RF frequencies, then there should be no effect at all at RF frequencies by grounding it. However, you really should ground the cage for safety reasons. A cage is useful if you can test things inside it and often this means running AC power through an RF filter located at the cage wall. In this case, grounding is absolutely essential to safety. The AC line filter that becomes part of the cage has connections, usually with capacitors, between AC and the cage wall.
The amount of isolation that you get from your cage between the inside and the outside will depend on the frequency band you are building it for, and the quality of your construction. There is a limit to how good it will be with a single wall construction that will depend mostly on how you close the seams in the screen connections. If your frequency range is modest, like say for example 50 to 200 MHz, then you will do pretty well. However, it gets tougher to get good isolation as the frequency goes up. I have built a few at 800 MHz with single wall construction and a door, using copper sheet, and we only got about 40 dB of isolation even after considerable fussing with door seams and such.
Many commercially available screen rooms use double wall construction, as Tony suggests, because they are going for a very broad frequency range and upwards of 80 to 100 dB of isolation. You likely don't need that much, do you?