You are on the right track there ... Thats what I would do!
Problem is caused by humid air condensing on the cold surface of the cistern so you must either remove the humid air or stop the surface of the cistern being cold or accept the problem and catch the drips.
Here's some good and bad ideas
Warming the water up - Not an option in my book due to risk of legionella but I think there was mention of a 'device' on here a few months back.
Making a 'cistern cosy' - Many moons ago I went to an house where this old granny knitted such a 'garment' and it worked in that instance!
Boxing the cistern in - Not always an option
Accepting the problem and fit a drip tray - I recently saw what looked like a proprietary drip tray fitted below a cistern. Looked totally naff though.
Dont create airborne moisture - Not having a bath or shower isn't an option unless you are a minger
Opening a window - Helps, but what you need is
cross ventilation ie air being dragged in through the door and out through an opening so an extractor will help more but only if its in the right place. Fit it next to an open window and it will short circuit the air leaving the moisture in the room to condense on a cold surface.
Hotter air absorbs more moisture so when using a bath or shower turn the heating up a little. I know you will be chucking valuable heat away as well but that cant be helped.
I was taught that the general remedy for all condensation problems is dry heat and ventilation. It is now apparent that in a lot of cases that insulation could be added to that remedy as well.