Is your plant a Fishtail Palm (Caryota mitis - or a different species of Caryota)? If it is, it requires bright, indirect light such as a curtain-filtered south window, or (at least 400 foot candles of artificial light).
The soil must be kept thoroughly moist, but never soggy wet or the stems could rot. Never let the pot sit in the drainage water. Water the plant as soon as the surface of the soil approaches dryness.
Give the plant nightime temperatures of 65 to 70 °F and daytime temperatures between 75 to 85°F.
Fertilize the plant with a balanced plant food (20-20-20 - for example) once a month durinjg the growing season (spring to fall). Never fertilize a plant while it is in the dormant period.
Repot the plant in spring if necessary (they prefer to be on the crowded side) in a one size larger pot. A good soil mix consists of 1 part loam, 1 part peat moss or leaf mold, and one part perlite. Add 2 tablespoons of bone meal to 1 gallon of the above mix for good root development. Most commercially packaged potting soils will also be fine.
Young plants can be propagated at any time of the year from the young suckers which develop at the base of the plant. Seeds can also be purchased to propagate new plants.
Keeping the air moist where the plant is located will help prevent spider mite infestations.