Would insulating a basement prevent it from passively cooling the house in the summer?
Yes, insulating the unfinished basement would limit the cooling effect in summer and increase your cost of air conditioning. On the other hand, the uninsulated basement cools your house in the winter. The question is, which one would be less expensive if you want to keep your house comfortable in both summer and winter?

The only way of avoiding the tradeoff is likely impractical, you would need to build in an extensive air plenum behind the insulation which would allow you to extract cooling when desired by blowing air through the plenum. This is actually a form of geothermal energy.