What is best caulk for a 3/8" wide X 1" deep gap between a shower floor and wall?
I'd grout it. Then when the grout fails I'd do it again. The third or more time gets caulk as a last resort before gutting it.

Before the picture was up I had envisioned where the floor meets a tub; grout always fails there after a few years, but by then it's all pretty well settled and a touch-up grouting usually lasts much longer. 

However what you have there... erm, what do you have there? Any idea if that was a kit or home made? Is there a watertight, factory manufactured basin down there or what? I'm going to assume there is and someone added walls and a floor for aesthetics. Otherwise,  gut it! So, it kinda doesn't matter what you use, until it starts growing mold. This is where the breathability of grout wins, over the moisture trapping capabilities of caulk.

If you MUST use caulk, use silicone. Latex caulk gets grubby much faster and is otherwise unsuitable in a wet location; insure that your silicone is suitable. I personally don't agree with the answers there about always using caulk at changes of plain; I try to get away with grout for as long as possible, everywhere. You can always come back and hack-job some caulk at it. When the time comes to readdress it, I'd rather be dealing with lose chunks of grout than removing old, moldy caulk mess. And if you use caulk, you will be readdressing it at some point, instead of the maybe you'd get with grout.