I had this same problem and set out to fix it. The first thing I tried was using a waterproofing spray (silicone) made for tent material. I gave it several thorough coatings and used an entire can on the curtain. After re-installing, it was slightly better but I still had water drops coming through the material and getting on the floor, counter, etc.
The only way you are going to fix it is by installing a plastic liner. I went to Walmart and bought the clear vinyl shower curtain for $3. Remove the screws holding the curtain track to the ceiling and slide the clips off the rail. Lay the fabric curtain down on the floor and then lay the vinyl curtain on top of it. You'll notice the curtain isn't exactly square, so don't try to measure length and width and cut the vinyl based on that. Just trace it or cut off the excess vinyl with the two matched up.
Next comes the tedious part... You have to attach the vinyl to the fabric. I considered glue, sewing, attaching new clips, etc. I found the easiest and most effective way is to mark the vinyl curtain at the top (where it's doubled up for strength) where each square clip is located. Carefully cut a square with scissors the exact same size as the square clip, and then push the clip through the hole so that the fabric and vinyl are now able to slide back onto that rail. The vinyl is now locked onto the fabric because of the rail system and won't come off, it's easy to remove if needed, and since it's clear it doesn't make the shower area any darker when showering at night.
I did this mod to mine, and along with corner splash guards installed and sealed with silicone, the water stays in the shower enclosure and my bathroom stays 100% dry. Cheap and very effective solution!