Rose varieties and the problems that plague each one are more numerous than can be counted, but there are a handful of common rose problems that will cause rose branches, also called canes, to turn brown. Unfortunately, by the time the cane is brown, that usually indicates that it is dead and should be removed. The exception to this is winter dormancy, after which brown canes usually flush out in spring with new life.
Brown canker is a fungus that enters the rose stems through damaged areas. Canes usually do not brown and die the first year of infection, but by the second year the damage girds the cane and kills it. Verticillium wilt starts on the lower branches and moves up the plant, giving the canes purplish-black streaks before they wilt and brown from the tips down to the base. This fungus persists in the soil for many years, infecting other plants.
Rose stems naturally die when temperatures drop low enough to kill the branch. Actual death usually occurs in spring when the plant starts to come out of dormancy only to have a late frost kill the tender branch. If the bush produces new branches with new growth from the base or only from certain canes, those that are still evenly brown and do not flush out are dead. Scratching a small area of the cane will expose either green stem underneath if it is still alive but late to flush, or you will see tan/brown flesh if it is dead.
Canes that are broken are usually obvious -- they are bent at an awkward angle or fall off entirely. Sometimes they don't break entirely or are broken in inconspicuous places. The length of the stem must be examined for damage, though the area where the stem changes from green to brown is the best place to start. Signs of gnawing or areas where the bark has been removed indicate gnawing rodents such as mice or voles.
Canker must be removed with sharp, disinfected pruners well below the canker spot and then burned or hot-composted. Verticillium wilt can be introduced to the garden from seemingly healthy plants. Purchase only from reputable growers and destroy infected plants. Covering roses during winter in areas where the canes are only exposed to brief bouts of killing frosts may ensure more canes survive the winter. Placing physical barriers around roses where dogs, people or equipment damage the roses helps, while spraying the base of the rose bush with rodent repellent discourages gnawing.