One of the houses we worked on was built in 1952 and was just listed. The Owner doesn't live there and called us after she got feedback that there is a musty smell in the basement.
A lot of times, when there's that type of smell, we see carpeting/pad that is or has gotten damp. But this basement has no carpet. It is not leaking either. But there is a smell.
Here is what we did to help the situation:
We opened the windows to air it out and turned up the air conditioning to make the house cooler and not so humid.
Another solution that we used was to install a dehumidifier. We bought the type that you don't have to empty (since no one is there) and drains automatically. That was about $200, but there are cheaper ones....especially if you can drain it.
Then we purchased a product called "Filter Fresh Whole House Air Freshener". We bought it at Home Depot. It's a great product that you attach to your furnace filter. It comes in 14 scents and the smell goes through the entire house. This product contains an odor conteractant that works with the fragrance to help eliminate odors. It can tend to be a little strong, but read the directions and you can cut the fragrance down. Filter Fresh is about $3.00, so it's a great inexpensive way to avoid smells in the home.
The items I listed greatly improved the smell and we haven't heard anymore negative feedback.
If this doesn't work, next steps would be to:
Mopping all the concrete with a bleach mixture and then rising the concrete.
Checking all the landscaping around the house to make sure the ground slopes away from the house and there's no water being trapped.
Sealing any cracks with concrete waterproofing sealer.