Remove the old caulk from between the tiles. Insert a utility knife with a fresh blade at the top of a row of tiles and cut downward in thin, short strokes. Take care to cut through the caulk as close to the edges of the tiles as possible without actually scraping or cutting the tiles. Work slowly to avoid scratching the plastic. Pull the loosened and cut caulk free with your fingers.
Apply rubbing alcohol to the edges of the tiles with a sponge to help soften and remove what is left of the caulk. Allow the rubbing alcohol to sit on the tiles for a few minutes, then rub vigorously at them with the sponge until the caulk softens and releases. Wear gloves while handling the alcohol.
Rinse the tiles with clean water and let them dry completely.
Squeeze a thin line of clear caulk between the tiles to seal them against moisture. Run your finger over the caulk lightly to help press it gently against the edges of the plastic tiles, sealing them in completely. Allow the new caulk to dry for at least 24 hours before using the shower.