This randomized phase III trial studies how well radiation therapy, paclitaxel, and carboplatin with or without trastuzumab work in treating patients with esophageal cancer. Radiation therapy uses high-energy x-rays to kill tumor cells. Drugs used in chemotherapy, such as paclitaxel and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells or by stopping them from dividing. Monoclonal antibodies, such as trastuzumab, can block tumor growth in different ways. Some block the ability of tumor cells to grow and spread. Others find tumor cells and help kill them or carry tumor-killing substances to them. It is not yet known whether giving radiation therapy and combination chemotherapy together with or without trastuzumab is more effective in treating esophageal cancer.