We have a traditional style tile tub circa the 1950s which has begun to
leak into the basement when taking a shower.  I have verified that it
is neither the drain nor water pipes and we have recaulked around
the seam where the tub meets the tile.  The grout around some tiles is
less than perfect and there are several built in soap holders. 
What is the best approach to solving this problem, hopefully short of
removing all the tiles and regrouting the entire tub?

I had a similar problem that I eventually traced to two sources:

1: Was the seal between the bath and the tiles. This was resolved by
removing all the existing sealant (which had broken down) and
replacing with new. (Note: Special Bath Sealant, not grout.)

2: I am also very splashy whether in the bath or shower (I am over 6
ft and I blame my height rather than just clumsiness). This I have
resolved by replacing the seals between the bath and the floor AND by
always cleaning up any splashes on the floor after every use.

PB