You might have difficulty empaneling juries if this were a requirement. What makes a good jury goes well beyond racial or sexual makeup. Also, in many places the jury pools don't match the community at large. This approach would therefore have the effect of jeopardizing the right to jury trial without really alleviating injustice.
A more fruitful approach might be to try to make the pools more balanced. Getting more community participation in the justice system seems like a very good thing at multiple levels.
I think we'd do better to, at least at first, start solving some of the reasons why our jury demographics are so skewed. Stronger protections for workers that need time off (there should be no work hardship exemption and employers should be forced to give jurors full pay for the duration of the trial) would be a good start to not discriminate against poorer people who can't take off from work without struggling financially or getting fired. Also, criminal convictions shouldn't disqualify someone after they've served their sentences. The current system disqualifies a disproportionate number of minorities and that leaves predominantly-white juries to convict even more minorities...a self-reinforcing cycle. Also, the SCotUS needs to recommit to the Batson doctrine to prevent prosecutors from dismissing minority jurors based on race.
A more fruitful approach might be to try to make the pools more balanced. Getting more community participation in the justice system seems like a very good thing at multiple levels.