How do I maintain air quality in a newly finished basement of a 1925 home with no ductwork?
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.