If the stem of an orchid starts to turn brown while the orchid is still in bloom, or before it has produced blooms from its buds, the plant is likely suffering from a disease of some kind and may not survive. Fungal diseases in the soil around the orchid can invade it through the roots and cause the stem to become brown as infection spreads. Treat the orchid with a fungicide to attempt to save it.
If the stem of an orchid turns brown after the flower has bloomed, the orchid is not dead. Most orchids bloom once a year and then return to a dormant state until the following growing season -- disposing of an orchid after it sheds its blooms is an easy way to miss out on years of continued flowering. After the blossom drops, the stem of the orchid and any remaining leaves will slowly turn yellow and brown.
The browning of the stem and leaves is not indicative of death; rather, the roots of the plants are taking in the remaining nutrients from them for sustenance during the dormant period. Removing the brown stem or leaves too early can weaken the plant for the following year and reduce its ability to blossom. Allow the stem and leaves to become totally brown in the planter before trimming them down. Leave some of the stem above the soil surface if it is a phalaenopsis orchid, as it will flower on the same stem.
After an orchid has developed the brown stem, storing the planter until the following growing season can present a problem. An empty pot with a small brown stem is not an attractive centerpiece for an indoor flower display. Keep the orchid in a cool, dark, dry location until the following spring when the stem will begin to produce growth again as it is exposed to light, fertilizer and water. Keep the planter dry to prevent the growth of harmful mildew.