The easy smart mouth answer: stop using it as a shower.

The harder but correct answer: tear it out and start over.

The definitely more expensive and difficult solution that doesn't involve demo: FInd an experienced solid surface fabricator/artist to cut a sheet of Corian to go over the tub deck to make it an undermount. With heat, the solid surface can have a bit of a curve bent into it to redirect the water back into the tub. You'd also need the walls clad in solid surface and solvent welded to the tub deck to keep all the water going where it needed to go.

Wow, thanks for the (brutal) answer.
Is there something else other than Corian that can help me?
I mean, building like a mini tile wall would help at all?

You could get a shower rod system for a freestanding tub shower and hang a curtain all the way around... This is only really a temporary solution. Live-wire is correct though. That tub should never have been a shower, and I'd worry about your structural support under it (unless it is on slab).

Here is a link that might be useful: something like this

When you shower, do you place the curtain inside the tub? Your shower curtain should be 2 layers. The outside red you shown in your photo and a clear plastic one on the inside. When you shower the clear plastic one gets pulled into the tub. There should be no water leaking out this way.

elphaba: thank, keep me posted

kirkhall: what is a slab?

chispa: it doesn't matter the curtain. Water comes from the walls to the sides of the tub, down the floor.

Here are pictures of the tub inside the alcove. I think he expects to start tiling the walls with subway before the end of the week. If not then, then soon. The tile will encroach on the tub lip even more - maybe 3/8 inch or so, I think.

I will have a small shelf instead of niche at the front o the tub down below the hand held shower rod and head.
Note that the kerbi wall board is on the lip - not much deck but at least I will have the shelf. Pony wall shown in bottom left part of picture.

Below is the back of the tub - same small lip. Also, in bottom right corner is part of the opening where the shower door will go. Notice there is no deck to sit the door on top of. That is a straight downward "cliff" to the floor where lip is totally covering the deck. Shower door will have to "sit" on top of the lip.

The tub is a small 48X32 inch drop in
Greek Kohler Tub.

p.s. there is another method that you can find info about that requires the addition/installation of a tile flange along the walls. My contractor has said that he thinks the kerdi along with the way he will deal with the seam between the kerdi and the lip of the tub provides better waterproofing than the flange. I'll hopefully get pictures of that too. My contractor has said this method is not supported/recommended by Kohler or Kerdi. It is totally non-standard. My only other option besides this tub was an American Standard tub that is very shallow and probably most often used in hotels. I'm so glad to have the option of this tub.

I mean, building like a mini tile wall would help at all?

You could furr out all of the walls so they overlap the existing tub "deck".

The thing is, it's a drop-in. So before going that route you want to make sure that any water that hits the existing flat deck on the rim of the tub would actually flow into the tub and not pool against the newly furred out and tiled walls.

Furring out the plumbing wall, you'd have to pull your plumbing off the old wall and install it in the new wall.

Honestly?

With that, or any of the proposed solutions, you're looking at cobbled together solutions. For a good amount of expense and effort you won't be getting a look you want. You're getting what you're getting. A cobbled together solution.

I recommend what kirkhall and others have written, simply getting and installing a 360-degree shower rod and a few overlapped shower curtains, the type of setup typically used for clawfoot tubs. Use it until it's time to remodel the whole thing properly.

How is this working out these days? I just installed a kohler Greek soaking tub/shower combo as well... Not a lot out there but I feel like I found a solution and am loving mine so far!