The amount of formaldehyde in insulation is far less than it used to be and some insulations are made formaldehyde-free.
You can buy a do-it-yourself test kit (google formaldehyde test kit) for $75 or $100 that will give you an approximation of your concentration, if any.
- If it does turn out to be an issue, ventilation is the only solution - lots of it.
- Open any windows or outside accesses and put in a fan that blows OUT of the basement.
As far as your question about placing another plastic barrier on the inside of the studs - no, don't do it.With a 6 mil barrier on the outside, any moisture that gets into the wall, up through the bottom sill or just from any puncture, must dry to the inside and can't if there is an interior membrane.
In a basement such as yours, humidity control is very important. If you don't have a forced air supply going into the basement, I suggest you install a humidity sensor to monitor the humidity level. Keep it under 70%, even if you have to use a dehumidifier.
Lastly, radon is a significant concern in some areas of the Country.
- Find a radon map of your area or ask around to see if its a concern where you are.
- If so, special ventilation techniques are required.
For more information:
Read "Does insulation containing formaldehyde continue to offgas into the home after the walls are sheetrocked and painted?" a Q&A answered by William Fry.