Using bathroom silicone sealant...

[HTFB]

The same trick works for shower trays. Ninety kg of storage-heater bricks have ensured that my version of that seal has stayed nicely in compression for three years or so, despite the innovative approach to fitting a shower tray without upstands that the moron employed.

[HTFB]

Scraping out the grout and replacing won't take *that* long, but silicone will work as long as you can get the area, preferably including the inside of the crack, dry. (Something like CT1 or Plumber's Gold will stick to a damp surface where silicone won't.) I'd be inclined to finish it flush with the tiles rather than trying to indent it like the current grout.

Achievement unlocked: grouting. They say life begins at 40.