Generally speaking, plants do not use their stems to breathe but there are always exceptions (like cacti). The rule is that plants respire through their stomata, which are (typically) located on the undersides of leaves.
edit/note: A qualification should be made regarding my overly generalistic statement above. There are lenticels on stems which means that stems do breathe. But as a matter of comparison, there are (say on average) about 300 stomata per square millimeter of leaf surface, whereas there are (say on average) 1 - 5 lenticels per square inch on the bark (also, there's a lot more surface area on leaves). So stomata are by far, the primary portals for respiration (compared with stems).
Plant stems are typically better off dry but there are many exceptions, as various plants are adapted to various environments.
Roots do need air (roots absorb oxygen primarily through root hairs) because the plant (mitochondria) needs oxygen for cellular/overall metabolism. Plants also metabolize sugars and require oxygen to do this. So, you might ask yourself, why not take/use the oxygen waste from the leaves... or why is the oxygen that they produce from photosynthesis in the leaves wasted or not used for metabolic purposes. Again I'm sure there are many exceptions to the rule but the answer is because (primarily) segregating/sequestering hydrogen and releasing oxygen (waste) from the immediate vicinity of the photosynthetic reduction of water helps to bias the chemical equilibrium in favor of the reduction.
AND:
Cellular respiration (via mitochondria) occurs everywhere in the plant (not just the leaves), and generally speaking, the fuel is commingcoming from the leaves while oxygen is commingcoming in through the roots (edit/note: oxygen is also commingcoming in through lenticles- located on stems, leaves, and roots). More specifically, sugars (primarily glucose), proteins and other organic molecules are commingcoming from the leaves. Transporting glucose and oxygen together from the leaves, would probably result in premature oxidation of the glucose (in the phloem). In other words, to successfully transport fuels, you don't pre-mix them with oxygen.