What does the "off odor" smell like as best as you can describe it?

For venting just leave the door open until you figure out the source of the smell.

It could be smelly shoes, a dead mouse or clothes hung damp and close together so they grew mold instead of drying.

I would try to treat the cause, rather than the symptoms, try what gammyt said, (check very carefully and clean it out as you go) but I'd also go up into the roof cavity and see if there's any evidence of the roof leaking, or condensation - this smell might be an indicator of one or the other, or a condensation problem. Also, is the roof cavity vented, with a whirlybird or similar?

Venting and/or slightly heating (but you have to be careful with the latter and ensure any heating complies with codes - you can't have a bare light bulb for example) might be a good thing for a closet anyway....you might consider a bathroom fan with a timer for example....and/or louvred doors if that's an option.

If you are absolutely sure there's no damp issue or dead critters (here's a thought, put an hygrometer (humidity meter) in there and see what it says, compared say to the rest of the house- you might try what I've been thinking of doing, putting some raw cedar panelling in there as an anti-bug and anti-smell treatment, I think I'll probably line the back of my closets this way.

I used to live in Sydney Australia, a damp warm coastal climate where mould etc was a real problem, I found stuff in closets would smell, eventually, even stuff that hadn't been worn at all.

The solution for me was to wash it all regularly - and thin it out (as in chuck stuff I wasn't wearing) - and to air things out, in the sun if possible.

I think walk in closets do start smelling after a while - it's the result of recently worn shoes, clothes that are worn more than once between washings, etc. I installed a small bathroom vent fan in the closet ceiling that vents to the attic (we live in So. Cal so there's no issue of blowing warm humid air into a cold attic - otherwise you could duct it to an opening) and have it on a timer switch. About once a week I turn on the fan for an hour and it seems to do the trick.