Adding additional insulation to the gap between the framing and the exterior wall can be a good idea, as it can help improve the energy efficiency of your basement and reduce heat loss. However, it's important to make sure that the additional insulation is installed correctly to avoid any potential downsides.

One potential downside of adding insulation to this gap is the risk of moisture buildup, which can lead to mold growth and other moisture-related problems. Moisture can accumulate in the cavity between the framing and the exterior wall, especially if there are any air leaks or if the insulation is not properly installed or ventilated. To avoid this, it's important to make sure that the insulation is properly sealed and that there is adequate ventilation in the cavity to allow any moisture to escape.

Another potential issue is that adding insulation to the gap may reduce the amount of space available for wiring or plumbing, which can make future renovations or repairs more difficult. If this is a concern, you may want to consider using thinner insulation or alternative insulation materials that take up less space.

Overall, adding insulation to the gap between the framing and the exterior wall can be a good idea, as long as it's done correctly and with proper attention to moisture management and ventilation. Consult with a professional contractor or insulation specialist to ensure that the insulation is installed correctly and to help you choose the best insulation material for your needs.