You are in a dilemma.
Soaking dry clean only drapes is liable to shrink them. I would offer to pay to have them dry cleaned. It might be cheaper in the long run.

On second thought, your landlord should have told you not to leave the curtains closed all the time. The water is probably from condensation on the window in winter. With them closed all the time it creates a cold zone behind them, but not as cold as the outside. Ice or just condensation on the window has no where to go and can't dry up because it is away from the heat. This can happen even with new airtight windows.

My experience is that bits of dust combine with the moisture leaves an impossible-to-remove stain from fabrics. You may want to risk dry-cleaning. All I can say is good luck, lets us know what you do.