You can spray a lot of febreze on it, but honestly the new drywall will cover the smell.

There could be many reasons in a 100 year old house. If it is over a basement or crawl space, stack effect air could be bringing up moistness/smell from the foundation walls or dirt floor. Multiple layers of siding over the years may have a moist sandwich. Subfloor could have had many years of pet pee soaked in. But old wood in itself that is covered up with new drywall agree shouldn't be a problem.

I would spray with vinegar and let dry.. Anything wood. Be sure not to let it blow into your eyes.

Thanks for all of the comments so far. I didn't want to damage the wood. I wondered about spraying the stud bays with a Kilz type of primer.

First find the reason why, or rate independently the level of what you consider mustiness, as one person's musty may be another's fragrance.

I would track down the area that the musty is coming from. Big suspect is the basement, if it has one.

It does have a basement.

Then, check out that basement good. If the musty is coming from there (more than likely it is) then, it wont be going away, when you put up new sheetrock either.

makes sense, thanks

Have you tried running a dehumidifier in the basement to see if that reduces the mustiness?

A dehumidifier might help, but, it depends upon why there is a musty odor. Does it leak, do walls feel damp? Is there any mold or mildew? If so, and its from leakage, dehumidifier wont help until the problem has been repaired.

Basement floor: is it concrete or dirt?

If dirt, installing a plastic vapor barrier, will make a difference. Have it installed professionally.

It's a concrete floor... an old Michigan basement, and yes, there is some leakage.

Might want to see if you can get it all cleaned and the leakage repaired. Otherwise, you will have that musty smell forever.

100 year old wood will decompose just as new wood will when moisture is present. with the presence of moisture just from humidity it will grow bacteria, mold and mild dew which will produce toxins, and this is what causes the musty odor. The wood provides ample food for the microorganisms but with limited moisture the population is very low so decomposition is very slow over centuries, but the wood still will eventually rot over time.

To eliminate the musty odor you need to maintain the lumber in a low humidity environment, and something which is nearly possible. The alternative is to use a biocide to kill microorganisms and when they sprout from a spore. A copper based product will do the job and will continue for decades. Another option is boric acid, or borax mixed in water or glycol sprayed on the wood which would kill the bugs for about 6 years. The spray needs to penetrate the grains.

before they install the sheet rock inspect the moisture barrier (plastic sheeting) and verify all holes are taped with the right tape and all seems are taped. This will create a barrier between you and the smell. The plastic should be on the ceiling and walls with all outlets sealed with tape. Concerning the floor I have no clue what you can do, but it's a good place to spray a copper based product on all exposed wood.

Pretty much always musty indicates moisture. Eliminate the water and the small will disappear. A dehumidifier (or three) in the basement is a real good first step. Get some hygromeers and get the air in the basement down to ~40% relative humidity. I'll bet you see a marked improvement within a month or two.

Check your main drain pipe. If is backed up it will give off a weird musty smell. In an old house, it could be the reason.

Concrete basement and drain pipes sound most likely if your home is completely empty and down to the studs. Now is a good time to check for plumbing leaks. A plumber also can advise you on cleaning out and inspecting all your drains. Where I live they can run a camera into the drains and check for tree roots. 

Your basement floor leaks? What do you mean? Seepage through walls or from floor? Some people where I live have seepage through walls and have the ground around the foundation removed and install drains and a moisture barrier. They also have moisture barriers installed indoors on floors and walls. Do you have a specialist in your area? 

While you are at it, have you checked your basement for radon? That's a serious concern too in the Midwest. I lived in an old house once with cracks and holes in the basement. When the next owners closed those they suddenly had radon in the house.

Radon test was negative, thank goodness.  There's leakage seeping through where the water enters the basement from the well. Not sure if it's ground water or if the plumbing itself is malfunctioning.  The well pressure is normal, so that might rule out the plumbing as being the culprit? There is a portion of the house on the leaking side that has zero gutters/downspouts. Hoping once this is remedied that the leakage stops.

To me both or these or a combo is where you musty smell is coming from.

@bliss1417 water is a potentially serious issue. So you think all the water is from a poor gutter setup? That would be wonderful. Do you think it will solve the musty smell issue too? Very pleased to hear radon isn't a problem.☺️

Thanks for all the great suggestions. I will get back about the outcome. I'll definitely have the gutters taken care of and the drain pipe scoped with a camera.