Nothing is going to cave in. What you need to do is get a length of 4x4 or 4x6 (even better), some bottle jacks like the ones under there now, and some 12" pieces of 2x12 to put under the jacks to distribut the weight, and the length of 4x6 across the underside of the joists that need lifting.
What is under there now? Are the bottle jacks that you see directly supporting the joists, or is there a beam down the center?
Ideally, you would either use the existing beam or put one there if there isn't one, pour footings and place posts about every 4 to 6 feet. I think I can safely assume the space isn't heated, in which case the footings should be set below the frost line. Doing it right is no small task.