The tile and grout in an all tile shower are not waterproofing materials but long wearing easy cleaning finish ones. IF your shower was built properly it goes like this from top to bottom...
-The tile and grout laid in thin-set tile cement.
-A layer of portland cement or thin-set cement
-A thick rubber waterproofing membrane
-A layer of portland cement or thin-set cement
-Plywood subfloor
The two layers of cement with the waterproofing membrane sandwiched between them is what does all the work of carrying the water from your shower to the drain and keeping it out of the rest of your house. Hopefully this was done properly...It can be tough to know. If you're lucky and you somehow have easy access it is worth inspecting the subfloor and joists underneath the shower. If I inherited such a shower that was having problems I'd cut a small hole in the sheetrock or plaster from the ceiling beneath the shower if it was in the way and I'd have a look or I wouldn't be able to sleep at night. A leaky shower of this kind probably won't pour water all over the place (unless it's really bad...) but let it weep and drip slowly, silently doing damage. The only way to know for sure is to look. I think this should be done first before anything else so you know what you're working with before doing any repair work. If it's anything but dry under that shower you'll sadly want to be redoing the whole thing...or leave it and never use it again...
If it is nice and dry, you should be good to go to replace the grout. It's tough to say why grout fails, as many factors are at work: the quality of the grout, the consistency it was mixed to, the humidity and temperature on the day it was laid in, how well it was laid in, the mineral content and pH of the shower water, and the list goes on and on.
The surest way to repair grout is to replace it. This involves scraping it out by hand first. There are different shaped diamond tipped hand tools for just this operation that ease the process a bit. You want to get absolutely all of the crumbly stuff out of there, and then even sound grout you'll want to gouge out so it's at least half the thickness of the thickness of the tile, and more removed is better. A strong vacuum is a handy tool to have for this process so you can keep the area you're working on free of dust and see what you're doing. Once you're satisfied you can simply mix up some brand new grout and lay it in there. A fine grit grout should be used.
Though there is no complaint of loose or broken tiles in your question, another possibility with these showers is that the structural system (the joists and subfloor) are not doing their job properly and letting the tile and everything beneath them to the subfloor flex, causing cracks. I've seen a shower like this built atop a plywood subfloor whose joints did not land on floor joists (...) so the whole thing was unstable and had to be ripped out and redone after stabilizing the subfloor.
Good luck. These showers can be monsters...hopefully yours was built well and just needs the cosmetic refinishing of the grout.
Well this is working just like a travertine shower should.
Travertine is porous. You can apply sealer but honestly unless the travertine was honed specifically to take sealer - which most residential tile is not - sealer may help a bit but in no way will seal your tile shut. Trust me I have been there.
So water is getting into the tile and the grout probably and will travel the course that gravity tells it to. It is perfectly normal to see water coming out of grout in a shower floor - especially on a porous stone.
So why could this be an issue:
You do not have proper backing behind the travertine. Meaning you do not have some sort of concrete backer or waterproof membrane. If you have drywall behind the travertine you would basically need to paint on the sealer every other week and that might not help. I seriously doubt anyone installed travertine on drywall but you never know. The only way you can tell is open up the wall behind it. Even then you always install drywall+plastic+thin backerboard which works - but would make you think it was installed on drywall.
If the water is making it outside of the shower area from below. You are on a slab which takes away 95% of the issues. But you still don't want water spilling to the next room. If the water is contained in the shower... well that is what the system is supposed to do.
A properly designed shower system isn't supposed to keep water from going in grout, it is supposed to last and keep water in the shower. If it is doing those two things it is doing its job.
What should you do?
Look for water in surrounding rooms - you don't have to go all Mike Holmes here. I would personally just point the shower at this wall maybe 3-4 times for 20 mins in one day. Let it get really really wet. And then see if that day or the next there are any moisture issues in the next room - you should be able to put your hand on the wall and feel this or pull up a bit of trim or carpet. You do not need to open walls.
If you do find water you need to open walls though. See what is behind the travertine. It is usually at this point where you have to seal it from behind or figure it was made wrong and start over.
However the most likely scenario is you don't need to do anything at all and let the shower function. I would not under any circumstance caulk the bottom. You know the water is getting in the wall - why would you want to hold it at the bottom? If you feel the need that you HAVE to caulk the bottom leave drainage gaps in the caulk. I would personally not seal the travertine if I thought (and I would assume this until proven otherwise) there was proper waterproofing behind. The reasoning is that with something like travertine you are never going to seal it perfectly. So water will get in, but if sealed well the water may take much longer to leave/evaporate. There is a good chance that you adding sealer already has caused more water at the floor because it is not evaporating and it can escape there. Nothing to be alarmed about concerning past sealing - just telling you why you are getting the result you are.
Best Answer
The best way to confirm if your shower basin has leaks around the tile is to plug the drain. Remove the grate and install a stopper of the appropriate diameter. It must be 100% plugged. Next fill the basin with water. It is easier to use 5 gallon buckets filled from another water source. Fill the basin to just below the curb. Mark the water height with tape and/or marker.
Wait awhile and check if the water level has dropped. If it hasn't let it stay over night and recheck. If no noticeable change in water level there are no leaks in the grout or joints.
I would suggest re-grouting those joints that have missing grout and any others that have loose grout.