The moisture laden warm air in the bathroom (you will always have moisture laden warm air in a bathroom) is condensing on the cold cistern - you knew this.
Provide more ventilation if you can - open a window a bit especially after bathing. Let the damp air out. Consider a vent fan in a wall or window.
Drape a towel over the cistern to keep the moist air away from it. Or find a way of warming the cistern up - even a little bit. Crank up the radiator.
Check the overflow pipe - if cold air is blowing directly up the pipe the large body of water in the cistern can become very cold - I had a tank in the loft freeze up because of cold air up the overflow. Get an angled bend on the pipe to stop the direct blow.
Clean the staining on the floor and put down bath mats to catch any condensation that might drip off.
On a personal note: Cover up the wooden floorboards and put down modern vinyl - the boards were never intended to be left exposed and look quite ugly with their gaps, dents and nail holes - splinters too.
Peccavi
November 2008