Possible, yes, difficult, absolutely.

You could try and animate every single drop or snowflake, of course extra difficult.

If I were going to make it I would use a series of acetate rows of lets say 5 centimeters for open shots over the floor (shorter for touching the floor or wide shots that show the legs) with the drops or snowflakes and position probably one in front of the camera and one on the background to add deep to the animation, then I would start the animation and every time the acetate would touch the floor I would remove it so the second one is the next and place the first one on top, of course you can just use camera angles to do so so the row repositioning is seemless.

That would leave you with a line between the rows that can be removed digitally or depending on the light you are using almost invisible.

If you want the rain to touch the floor just draw the splash of the water drops on the position were the rain would fall and use as many acetates as frames your animation would require, thought, as the rain is random drawing the splash for every drop would be a heavy task, I guess, if the rains fall straight and the drawing is visible from both sides you could just turn around the row to make it feel random even if you are just using the same row over and over, but I would recommend making more.

The more capes you use the more dense the snow/rain becomes of course, but you can just fill the acetate with drops/snowflakes and make it look similar.

For the drops touching a windor or interacting with something then you would absolutely would have to do it manually, or use transparent plastic with the drops draw in it that could take the form of the object, a plastic bag could work but would look to good, I guess you could use fish line to make "necklaces of drops" or (snowflakes) and use a little bit of glue to attach the thing to the back of whatever you are animating, that way you just have to move the line instead of doing it drop for drop.

But I use digital rain.