Nothing will help other than windows when the the wind is blowing the rain around

Get gutters. It makes a big difference.

Lanai rain at bottom
 

We live in a patio villa and had the same problem. Grass trimming constantly chewed up the screen, as well.
Go to White Aluminum on 441 in Leesburg, on right, just after the 27/441 split, and get rolled white 20 inch rolled sheet aluminum (measure what you need, plus 6 or 8 inches for good measure). It comes in pebbled finish or smooth (we like the pebbled). Get the edging channel to hold the aluminum by picture framing the sheet aluminum and plenty of self-tapping screws. Fairly easy, as long as you have some tin-snips, ruler, pencil and a helper.
White Aluminum is an installer of lanai and porch enclosures and additions. They can do it, if needed or any competent handyman. Just takes about 2 -3 hours or so.

Lord Arc has decreed
 

finish thy pint and get thee to the stocks

. I had "screen drains" installed and (for me) a total waste of money. The fact they aren't completely flush to the cement "floor" makes them useless for other than A NOAH like flood! I got them mainly for when I power washed the floor so the water would flow out ..... NOPE, it just sits under the slight height above the concrete ground those drains sit and I still have to sweep out the water into the screen drain instead out of the screen DOOR. Even if they sat directly on the floor there would still be a lip to stop it from draining fully or close to fully. Fortunately, they weren't too expensive. That was my experience, YMMV.

Lanai
 

First get gutters to redirect roof distribution of rainwater, then invest in motorized screens to protect your furniture in the lanai, rain will blow in the lanai from every direction. Why fool around, just do it right the first time, then enjoy Florida living!!

The resale we purchased had thin plexiglass on the bottom of the screen to the chair rail. It’s about 14”.

Had clips to remove to clean screens. Kept the rain splashing on the rock bed into the lanai. Don’t know how long it was there when we bought the house in “2014” but it’s still there today, can’t see it from the back, still very clear with few scratches, with no yellowing, lanai faces south.
So have first hand experience, I am guessing you personally tried plexiglass and yours yellowed? How long did it take to yellow?

Up north our pool “fence” was thick plexiglass, so had a great view all around the deck. Through spring, summer, fall, and yes winter, it still looked great when we sold.

Really! IF you don't want rain in the lanai, enclose it. best solution! no reason to look like you came from the backwoods of ? well wherever. I've enclosed 2 lanais as my second house on Southern Oaks had lots of blowing rain... This is a personal decision, no one here can make that decision for you.
:pray:

That sounds attractive...not! Just wait until the trolls see that.

We added a garden bed with small shrubs and mulch directly behind our lanai. 20" deep. It effectively cut down on most of the splashback through the screen.

Rain into lanai
 

I hI had Todd's Screening put in vinyl glass all along the bottom of my lanai screen. It keeps the water out of the bottom so the water doesn't get into the floor of your lanai. To be honest the best way is just to enclose it. Nothing is going to keep out rain entirely.

I agree, but they said they don’t want to put a lot of money in their Lanai. I don’t know what to tell them, if they start tack things on it it’ll look terrible!

Our lanai is not enclosed. It faces ENE and sometimes get fierce rainstorms with wind. That kind of rain overloads the "super gutter." There are three 2" holes drilled in the concrete near the edge of the birdcage allowing the water to drain. The furniture is cast aluminum, the upholstery is Sunbrella. After experience living on a sailboat constantly exposed to the elements and under a bimini made of Sunbrella, we don't worry about it.

Hmmm, you must be new to Florida! :laugh: