My tack lives in the integral garage to keep it mould free. It's cold, but dry.
A dehumidifier extracts the water (cold and dry), whereas a heater evaporates the water AND makes the tackroom a nicer place to be for humans.
Not sure which is cheaper to run though - heater V dehumidifier
heater. afaik you don't want leather too dry, it'll crack. (white) mould is apparently okay on leather (although i hate to see it).
i have a plug-in radiator in mine, just move it around to where it's needed, good for drying rugs too, i stick it under them and the heat rises.
I ask because I've had a little electric radiator in there for the past 3 winters. It doesn't really seem to do any good and costs a lot to run. I thought a dehumidifer might be cheaper to run and take the damp out - it seems I might well be wrong.
You need to prepare the room for the winter, empty it out and clean thoroughly. Hang everything out in the sun to really dry out. Plug any holes in walls, etc - and loose fitting windows. On some of my windows I use paper towel, which I fold several times and then press into the gaps using a blunt knife.
To make cheap draft excluders for windows buy a sheet of very thin foam and a roll of double sided tape. Lay tape along a piece of tape then cut along the tape line, clean the window frame really well, scrub with meths, allow to dry and then remove the backing tape off the double sided tape and stick onto the frame.
Heat escapes through glass - if you do have windows double glaze them with clear plastic and masking tape. If you are happy for it to be premenant then use duct tape to secure the plastic. Clean the windows well and make sure the wood is dry.
Cheap door draft excluders cut a strip of carpet and nail to the inside of the door so that it lies tight against the floor and the frame. Try to put only dry things away in the tack room.
Hang damp numnahs over a washing line in an empty stable or under cover. Coat tack that is not likely to be used with a layer of vaseline or solid base leather dressing. Store inside a large plastic bag.
I have an oil filled radiator that I leave on low all the time, room may not feel exactly warm but there is never condensation on the window which tells me the room is dry.