The headline of your question is insulating an old house with no vapor barrier perhaps the first thing you need to know is that the presence or absence of an interior vapor barrier is irrelevant.
Insulating interior walls vapor barrier.
If you want a vapor retarder on the interior side of your walls you can always apply vapor retarder paint.
Vapor barriers when to use them when to not.
The place to control vapor migration is on the exterior of the basement wall in the form of waterproofing or damproofing.
During the energy crisis of the 1970s a prevailing belief took root where it was thought that tightly sealing walls and ceilings with a vapor barrier was essential to blocking heat transference and reducing energy costs it was soon determined however that unless the seal was absolute moisture that did get into sealed walls could create serious structural problems and health issues such as.
A vapor or moisture barrier is a layer of protection that is designed to act as a waterproof shield helping to keep condensation away from wood insulation and other mold prone materials that are located behind the drywall.
To know if you need a vapor retarder ask yourself a few basic questions.
Bathrooms and kitchens produce an enormous amount of water vapor daily.
In any case the vapor barrier must point to the warm side.
By installing a sheet of poly there you cut off that drying mechanism and water that finds its way into walls can stay there longer.
This barrier is meant to keep moisture from getting to the insulation in the walls and ceilings and it is required by building codes when insulating most houses.
Without poly beneath the drywall water vapor hits the drywall and diffuses through to the drier in summer indoor air.
Vapor barriers are sheets of plastic or other material placed on one side of insulation sheets.
On the interior the best insulation to place against the concrete or block is rigid foam without any facing such as foil or plastic sheeting.
A vapor retarder is a material used to prevent water vapor from diffusing into the wall ceiling or floor during the cold winter.
The presence of an interior vapor barrier makes drying out the cavity harder to do though.
The short answer is no you generally do not want a vapor barrier on the walls of a finished basement.