The primary reason to ventilate an unconditioned uninsulated attic space is to allow moisture to escape. Moisture moves like air pressure only very slowly so it doesn't take much of a vent opening to lower the humidity if the air outside is drier. If there are eave and ridge vents even less vent area is needed.

The problem with adding a fan to the attic to move more air through it is that it could draw humid air into the attic but that is highly dependent on your climate so I can't address that issue.

If you have a musty odor in the upper part of the house, the first thing to look for is a roof leak especially at the flashing of roof penetrations and possibly a bathroom exhaust fan that dumps or leaks moist air into the attic space.

It sometimes helps to use a vapor retarder paint on the ceiling of the bathroom if there is no exhaust fan. If the moisture is not entering the attic from below or through poorly constructed exhaust ducts, it is entering from above.

Ridge vents and the necessary soffit vents use physics to ventilate an attic space. The air in an attic is warmer than ambient(outside) air because it is warmed by the house and the sun.

Warm air is lighter than cooler air, so warm/hot air rises. The soffit vents admit cooler air to displace and allow the hotter air to rise.

The ridge vent is placed at the highest area of the roof. That means the hotter air has easy escape to the outside.

Adding anything---more roof vents, gable vents, or fans actually decreases the efficiency of the ridge vent system, which only needs Mother Nature to work no matter what the weather/heat/ conditions are.

The reason some folks think ridge vents need help is because the system---ridge vent and correct number/area/placement of soffit vents is incorrectly designed of the soffit vents are partially or fully plugged.

A musty odor can be caused by mold inside the walls, which could have been helped by a leaky roof over that room.

Replacing the roof may correct the leak, but will do little if nothing to correct the mold growth/smell.

If you stop the source of the moisture the fungus growth should stop.

Thanks for the information. We selected a roofer who did not advise a ridge vent. They are now in the middle of doing the roof and after removing all of the shingles we have been informed that some of the wood underneath is not in the best shape and the best thing to do would be to plywood over the entire front of the house (cape cod type home). His worker suggested putting in a ridge vent when I asked how the wood might have disintegrated to that degree. We did have a fire in this home 15 years ago and it was totally gutted. The roof rafters and wood were not replaced at that time though and it is the original wood from the 1950's that went through the fire and water damage. It will cost an extra $2000. Any ideas?

Damage to the roof sheathing is probably from leaks in the roofing rather than excessive moisture in the attic but without seeing it I can't say for sure. If you need better nailing base or the sheathing is not structurally sound add the plywood.

If you don't install a ridge vent make sure there is another way for moisture to escape hopefully not from a fan which might add moisture to the attic. Also make sure moisture isn't being dumped in the attic from bath fans, etc.

It would help a great deal to know what the climate is.

There should be more ventilation at the eaves than at the peak. So, hopefully, that's what your roofers realized. (Too many roofers are only too happy to sell you as many things as they can. Explains those idiotic whirlygigs on rooftops.)

Besides the sources of excessive attic moisture mentioned by the previous posters, you may have uninsulated leaking attic ducts and/or a poorly-sealed attic to living space interface.

Here is a link that might be useful: Crash Course in Roof Venting