Leather Jacket Has a Musty Smell?

I have two methods
If the jacket is clean (except for the smell) put the jacket in a good box and unwrap a load if Irish Spring soap bars (dozen). Lay the soap around the jacket. Box size needs some air space. Use a box the size you would use to give the jacket away as a gift). Place unwrapped bars around jacket. Add some inside the lining. Replace lid.
Put the jacket away for a month and when you remove it from the box the smell will be gone. If not completely, replace lid and wait a little longer.

You can still use the Irish Spring in the shower.

I had a vintage car that had been leaking water into the trunk. Strong smell. My brother claimed (the sports car restorer) my method would never work in the trunk of this car.. Needless to say I had to get the car into a dry garage, out of the rain (Seattle). Boy was he surprised when the Irish Spring worked beautifully.
If your jacket is soiled as well as musty I'd suggest you might try washing a dirty leather jacket from the thrift store first, using this method.
Measure the length of sleeves and perhaps length of back.

Use cold water, soap and wash jacket in washing machine. If jacket is soiled on cuffs, neck, etc, I always use Zout(laundry spot remover on soiled areas first). (I have washed dirty leathers by the dozen). I have not had to deal with mildew but by adding vinear at the end of the wash, and letting it soak over night, it will be the secret to killing the musty smell. Add vinegar if you can still smell the jacket when jacket is washed and you are ready to add the softener to the rinse..

Hold off on the softener and add vinegar first.

Have water in washing machine (pour a couple of gallons into wash with 2 cups of vinegar) if water is gone.. Swish jacket around by hand if necessary and let the jacket soak all night.
Let the vinegar water drain for the rinse cycle.
Add one to two cups of softener to rinse cycle.

I always add a load of fabric softener to the rinse. At least 1 cup, maybe two. Once the spin cycle is over following the rinse, have a strong hanger waiting. You may need a place for the jacket to drip dry. (Tub or concrete floor) Put jacket on hanger and let drip (some garments will almost run water and others not so much).

Once the jacket is finished dripping I make sure it it hanging well on the hanger. Check out the length of arms and legs with tape measure.
Drying time will take about three days. If your jacket dries stiff, do not panic.

When is it completely dry, crumple it in your hands and roll it around in towel. Wear it around the house and you will be amazed how it will restore itself.

If you do the trial run thrift garment first, select a dirty item from the thrift store and pre spot it with Zout or other pre spot removers.
I also pour on the liquid detergent onto the worst areas of garment.. You do risk a chance the item will shrink a bit. That is why you want to stretch it slightly before it dries.

Try taking measurements of the thrift store item before you place in wash. Measure length of sleeves or the leg of pants. That will giive you an ideas of how long the sleeves or legs should be before you leave it alone on the hanger to dry. Stretch it a little everywhere like you were shaping a sweater before letting it dry.

I decided to give this method a try when they came out with washable leathers. I thought, did they reinvent the cow? Not likely.