I just bought a house built in 1925 but it has just been professionally renovated, including finishing the basement as living space. The home has a boiler for radiator and baseboard heat. There is no duct work for ventilation nor AC. My question has to do with whether or not it would be advisable to install an air to air exchanger in the basement.
The first thing I can recommend is doing an air quality test on the space you are concerned about. Tests range from simple DIY kits to having a professional come out and measure the quality of the air and possible contaminates. Also, since this is a basement, other tests such as testing for Radon as @mikes mentions would be adviseable as well. Once the test results are in, then you will know what (if any) steps need to be done to improve the air quality.
In general, air quality is dependent on pollution, the circulation of the air, and the cleanliness of the air.
Air pollution can be reduced by eliminating sources of pollution or if they cannot be eliminated reduce the amount of pollution emission. For example, turning your heating system or other high emission devices can improve air quality.
The circulation of air is also important. Adequate ventilation is required for good air quality. This is usually accomplished by bringing in air from the outdoors. If you have windows in your space, make sure they can be opened as an open window brings in fresh air. Fans or other types of air movement devices also promote ventilation. Adding ventilation systems can also promote air quality but this can be costly.
Finally, there is the cleaninless of the air. Ther are several types of air cleaners out there, ranging from inexpensive to hospital type systems. The effectiveness of air cleaners are gauged by how much air it is moving through the system and the amount of contaminates it can collect. Obviously a cleaner that moves a lot of air through and collects very little contaminates would not be that effective.
Keep in mind air cleaner products do not remove Radon or other gas pollutants.
So, to answer your question about air quality:
On the plus side, a 1925 house tends to be fairly drafty. Not an ideal thing if you live in MN, but it can be considered a benefit as it typically means you have plenty of airflow in the home.
The main air quality issue in your basement will likely be mildew/mold. That will depend on your climate (is it humid?) and the quality and type of finishing that was done in the basement.
If it is a humid region, definitely invest in a quality dehumidifier. If you already have central air, and it can handle the extra space, definitely add the basement space to the HVAC system to help circulate. Having heat in the basement is good too, as the warmer air will eventually rise through the house pulling in newer air into the basement space.
Other things you can do would be to install ceiling fans...a relatively cheap way to help increase air circulation.
And as others have stated, definitely get a Radon test.
I think the answer to this question is going to depend in large part on your personal opinion/experience, and how well the renovators chose their materials.
You should probably mount (at least temporarily) a humidity sensor in the basement, to support your personal observations.
Based on this information, you could elect to install either a whole-house air circulation/filtration system, or one just for the basement. This would essentially draw in outside air and vent stale inside air.