Those webs you've been seeing are a sure sign that your palms have been infested with spider mites. These almost-invisible pests suck plant juices and can cause a plant to go into decline. Give your palms a shower to wash off the mites. In fact, regular showers may be enough to keep the mites at bay. Palms are a favorite of spider mites, so you'll have to watch for their reappearance. Insecticidal soap, horticultural oil, or chemical houseplant miticides may be necessary to control the mites. Palms need more moisture indoors than they do outdoors. Keep the soil slightly moist, and place the plants in a bright, draft-free location that gets lots of indirect sun. Palms dislike direct sun.