I live in a 100 yr old house with a full basement which is a double brick wall with a 3-4 inch space between the two walls of brick.I do not get water, but it always smells damp and musty.
The basement ceiling has 3 inch paper faced fiberglass insulation (no moisture barrier). The entire basement floor is concrete except for two areas. One is a 5'x10' area at the back of the house below the bathroom which used to be an old coal bin and has a dirt floor-not a problem.This room is entirely below ground. The other is a 3'x5' extension off my dining room (bay window)also with a dirt floor beneath. This small area is at ground level. It has two windows directly across from each other- one opens to the outside and the other opens into the basement. Several years ago I covered the outside window with plywood. The parquet floor in my dining raised up. I went back out and drilled a bunch of holes in the plywood and placed a moisture barrier over the dirt crawl space and it helped.
I am getting ready to tear up the old floor in the dining room down to the original one. 25 years ago it wasn't in any condition to be refinished, so I'm going to put down engineered wood plank oak flooring. Once I get down to the original floor, can I first apply a 6 ml. moisture barrier with the underlayment on top of it? Or, do I have to remove the insulation in the basement ceiling, install plastic and put the insulation back up? This is going to be expensive and I don't want to screw it up!!

The paper facer on the fiberglass insulation is a vapor barrier. Is it against the warm side or the basement side? You absolutely don't want to have 2 vapor barriers. You will end up with some pretty weird moisture issues between the two if you install poly in the area where the paper vapor barrier is in place. The vapor barrier in your area should be on the warm in winter side of the insulation.

The fiberglass is against the warm side and the paper is against the cold side. I didn't install it, but I suspected it wasn't installed correctly.
My attic insulation was installed the same way, but it has those styrofoam vent sheets underneath against the backside of the roof. Is that installed properly? I don't need a moisture problem in my attic too!!

Many insulators install crawl space and basement ceiling insulation upside down, and when one tries to tell them they're wrong, they tell you they've been doing it that way for 25 years. In any New York region, your ceiling insulation is installed incorrectly.

In your attic, does the insulation follow the rafters all the way up? If so, the insulation is installed correctly, but the presence of the styrofoam baffles could be creating at least a partial second vapor retarder, and you may find that your insulation is wet up there, too. Those baffles are only necessary where the rafters meet the ceiling joists, for no more than maybe two feet, so that the ventilation path from the soffit vents. if any, is not blocked by the insulation being jammed into that tight spot.

To deal with your original question about the musty odors in your basement, every earth floor no matter how small should be covered with a vapor retarder. The brick foundation walls are likely to be allowing some moisture to pass through, but the biggest problem in older homes is the probable absence of a vapor retarder under the basement floor, and that is likely the biggest contributor to the musty odors.