1.   All this work will go for naught unless you ventilate that basement, to allow water vapor to escape.

2.   And your basement is ventilated.

3.   And, ventilate the basement in summer, which should be done anyway to prevent that enormous buildup of water vapor.

4.   Another way to vent that space is to open up the wall between the crawl space and the basement, and ventilate the basement.

5.   If that is what is happening, then ventilating the basement from April to November might help.

6.   Instead, ventilate the basement.

7.   It may be that the air exchanger you heard of was installed in the basement but does not ventilate the basement.

8.   Perhaps ventilating the basement in spring, summer, and fall would reduce the amount of water vapor there and thus possibly reduce the efflorescence.

9.   The only thing you can do to prevent further damage is to ventilate the basement in summer or use a dehumidifier.

10.   This will help ventilate the basement in the summer.