Industry standards call for caulking all changes of plane (i.e., corners, and floor-wall junctions). So, if there's grout there, you can buy a grout saw and remove it. It must be dry and clean of any mold before you can apply new caulk. then, it must be left to cure for the time indicated on the instructions of the caulk you select (anywhere from hours to multiple days). If the crack is deep, it is best to use foam backer rod prior to caulking. This does two things, lets you use less caulk, and makes that seam more flexible so it has a chance to last longer (the caulk ends up with an hour-glass shape because of the round rod behind it that stretches better than a big hunk).
But, for it to be weeping water for a week of non-use of the shower, it likely has other problems. Does the grout on the floor look like it is wet? How about part way up the walls? A leak behind the wall in the plumbing could cause what your'e seening, but not if the water pipes are on the other side of the shower. If the shower floor looks like the grout is damp, a more likely problem is that it was constructed improperly. Two things can cause this (that I know of): blocked weep holes in the drain, or a liner improperly installed flat on the floor. Many people think that the tile is the waterproof layer, but it is not. in a properly constructed shower, you should be able to not have leaks even before the tile are installed. In the pan, it is the liner that is the waterproof layer, and it MUST be sloped to the drain. Water WILL get under the tile, and it must flow down the slope of the liner to the weep holes inside the pan. If the liner is flat on the floor, water will (eventually) accumulate and create problems. It does not mean it will leak, but you'll start to get moisture problems in the shower since it can't dry out properly in between uses.