Much along the same lines as ED65, I suggest something like an oil based interior polyurethane varnish applied carefully with a brush: it is an interior door isn't it? There does seem to be some clear finish of some sort on the right hand portion already exposed, which is a little patchy, and I suspect a light sanding will remove the patchiness, perhaps using 150 or 180 grit paper. Oil based polyurethane varnish is tougher than the mixtures ED65 suggested, and a couple or three coats will leave a definite film, whereas his suggested options will give a thinner or lighter 'close to the wood look'.
Which finish you choose depends on things like appearance and resistance to abrasion and damage required. Slainte.