As a wine enthusiast... I'd be less concerned about paint fumes than I am about bottle vibration because it's under a set of stairs. How long do you store a bottle? If it's short term, NBD, but long term vibration can wreck a bottle. We have an under-stairs wine room and it's only for things we'll use within a year.

Use a durable paint (like a high quality BM or their Scuff-X line which we have used in those spaces!) and give it like... a solid week to cure and harden.

I agree not an ideal place for wine storage . As for the finish paint it and do not put wine in there until the paint cures usually about 30 dyas.

Look for zero-VOC paint. Don't use paints with any VOC. That's the bad stuff that can gas in. Also, after letting your paint dry for a few days, also let your room finish airing out for a week or two before you start stocking it.

If you have vibration concerns, maybe you can put in a vibration leveling system if you are currently in the planning stage. If you are dealing with more built in, you can help with that by using vibration deadening material. Most often rubber, neoprene, or felt, often found for industrial or acoustic purpouses.