One not so attractive idea (unless you get something that looks nice) would be to put a towel or something on the bottom. This would collect the water, get a couple and just cycle through them. It would stop the water pooling, and if you put some plastic down beneath it wouldn't even touch the wood.
My guess as to the problem is just the fact there is a box of glass on the outside of your house, so any moisture in there will immediately condense when it's cold enough on the outside, so any amount of insulation under the box / over the box ain't gonna help that matter. The clear plastic idea would stop the water condensing on the glass so much, but you'd have to make sure you got a decent seal around it, else it will just start to condense within the plastic (like when double glazing glass "blows").
Best solutions are to run a dehumidifier in that room, but then you're not really meant to run them when people are in the room too, especially with children. They are certain circumstances when you might need to, but they are generally designed for that purpose etc. Too dry an air isn't good when breathing it a lot.
The reason it will be better when your curtains are open is that it lets the heat through as well as the moisture so the glass warms slightly, thus not condensing as much of the moisture from the air.
Personally, without spending loads of ££, I would go with my first suggestion, get a couple of hand towels (or whatever size you need to catch the condensed water), and lay then on some plastic and rotate them through (can you tell I live in an old farm house with water features? lol)
It doesn't look amazing - but it doesn't look ugly either, although granted, a towel in a bathroom doesn't look out of place, any other room, it might. But it's effective nonetheless.
If it's causing any damp, you should be able to get onto your landlord to fix it, considering damp/mold (or is it mould? I never know..) is a health risk
Replace it with linux.
Doesn't the :) indicate that the post has been made in jest?
TBH the buttress makes it look more like a bodge job.
Drill a hole in the bottom to let the water out?
Take a sledge hammer to it early one morning, phone your landlord/lady and say you woke uo to a horrible noise abductions came down and saw that.
Where abouts are you, got a dehumidifier in the loft
My ps old house had a window like that, it had no support underneath and we used to sit in the bay as kids... Was always condensated
Yeap go with the polystyrene first as it will be insulation that is the problem, personally I'd go with Poly, building foil then maybe even glass batts before adding another bit of wood underneath it. Once that is done check the drain holes in the windows joinery are not blocked. May be forth trying to get some insulation in the bit above too if it has not been insulated.
If it is already double glased you may get a little extra from a third layer but given the thermal sink that is the lack of insulation of the rest of it, it would make no difference till the rest is fixed up a bit.
I think it will be much better to try and fix the problem rather than treat the symptom by burning through towels and power running all sorts of gear to simply alleviate and not solve the problem.