Be careful about buying into the NASA study hype. In the study, the plants were tested in carefully controlled lab settings that don't duplicate home environments well at all. All healthy plants provide some limited air cleaning ability, but only if they are healthy and growing vigorously. Instead of focusing on the specific plant species touted, select plants that will do the best in the environment you can provide. No ailing plant will provide air cleaning benefits. Overall, the best plants are the ones that are healthiest, growing vigorously and have the most total leaf surface. All of that said, unless your home is filled with healthy plants, the air cleaning by plants will be limited.
Those basket pots that you referenced are really not intended for standard plants used in the home. They may be appropriate for Orchids, some epiphytes and hydroponics, but not for the plants you cited.
If your plants are staying too wet, then it may be because the pots are too big, the soil is not sufficiently porous, the available light is inadequate or you are watering too frequently. There are no special pots or other devices that will correct those problems.