Sealing underneath stairs

[Telemichus]

We have an understairs cupboard, the stairs face the front door. The cupboard comes off the kitchen & I have been finding that cooking smells seem to be coming through the stairs into the ray of the house even if all doors closed.
the underside of the stairs is plain wood - do you think paint (gloss? Other?) and some decorators caulk would be enough to seal the underside a bit?
I had wondered about a sheet of white sticky vinyl lining paper stuck under. But obviously there are lots of fiddly bits o get round.

Sorry, any thoughts or advice welcome

Thr decorators caulk should do it I think.

I agree with pp. I think the smells will be going through gaps rather than through the wooden stairs themselves. So caulk in any cracks and holes.

Thanks I think caulk will stick better than The filler I tried before that just fell out into my eyes!

Get a powerful extractor fan for the kitchen, and use it. Shut the door and window. It will create suction in the kitchen that prevents steam and odours drifting out.

It will be more effective if it is mounted over the cooker. A wall fan will need to be more powerful.

Recirculating hoods are useless, except as ornaments.

Thanks @PigletJohn i have always wonder d about those hoods. We don’t have space for one anyway. It’s a ridiculous kitchen, but I will explore extractor fans.