Water seeping from grout in tile shower
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.