If you were going to regrout (you shouldn't, industry guidelines say not to!), you'd need to go at least 2/3'rds the depth of the tile so things had a chance to adhere well to the sides of the tile. If you can go a bit further, I'd use some backer rod in the gap. This does a couple of things:
- makes the caulk joint more of an hour-glass shape that stretches better rather than tearing off the sides (you still need to get those edges as clean as you can)
- means you'll use less caulk
Now, with that cove tile for the base, it might be okay to have that joint grouted...I'm not positive.
But, keep in mind, if you're doing this to help make things waterproof, it's a waste of time! Neither the tile nor the grout is waterproofing in a shower...the waterproofing in a pan is the liner. Yes, the vast majority of moisture will go sliding down the tile on top to the drain, but a little WILL get beneath it regardless of what you do. A properly built shower is built to handle that. That's why pans have a liner, and it should be flood tested prior to putting the tile in place.
It's not unusual for any change of plane to crack which is why the industry guidelines call for a movement joint (often a caulk joint, but could be a simple gap, or an engineered movement joint).
A grout saw can remove the grout, but it's harder to do in that gap against the wall. A vibrating tool with a grout blade MIGHT let you get in there, especially if you can find one with an offset blade. In either case, you have to be careful to not mess up the edges or surface of the tile if it slips, so go slow.
Sometimes, if the grout is not in great condition, a utility knife can work, but you'll need to replace the blade often.
Best place I've found for things tile related is
www.johnbridge.com