Usually wood can be stabilised, but it needs the damp driving out first, then I wouldn't go the varnish route myself, I'd use microporous woodstains or some of the better wood oils.
If it wasn't a posh door (or not very visible), then linseed oil and paraffin 50/50 works incredibly well (I've used that on wood outdoors). You just have to keep reapplying it but the great thing is that there isn't any laborious prep, just quick scrub off loose dirt and apply with a brush. The downside is it tends to go very dark, almost black.
If you use wood hardner, then you can't use the woodstain or oils (because they can't penetrate the wood). You are left with painting and despite what the manufacturers say, paint isn't impervious to water. Painted wood that is rained on will still swell.