I was wondering...this problem has only started to occur once the
weather here in Pennsylvania dropped so low. Is it possible that the
furnace is reacting to the cold in
the basement?

You may have to reset your furnace because the flame sensor
is shutting the furnace down because it senses no flame/heat.

The flame sensor may be a heat sensor called a thermocouple that is
put close to the flame and must sense the heat before furnace
operation continues. If you have this type of sensor the cold weather
could be effecting it if it was marginal to begin with. The sensor
looks like a bulb attached to a small copper pipe. It is the size and
kind of looks like a wooden match head. Two things you can do to
remedy a thermocouple problem.

1. Clean the bulb of any insulating oxidation.
2. Position it slightly closer to the flame.

The flame sensor may also be a light sensing device that must see a
visible flame in order to continue operation. It will have a lens or
see through glass window that faces the flame. If this is your type
of flame sensor then just clean the glass or whatever it is that faces
the flame.