Having a shoes that does not fit your foot correctly, especially your running shoes, can result in a lot of problems that can become difficult to deal with. If you purchase a pair of running shoes that are too wide for your feet and leave a lot of gap room, then you run the risk of developing painful blisters. If your shoes do not fit you snugly, and you do not have thick socks to help stop some of the rubbing, then the sweat your body generates from your runs coupled with too much room for your feet and ankles to slide back and forth within the shoe when you repeatedly pound the pavement during your stride, can result in blisters on your heels, toes, balls of the feet, and ankles.
Most shoes that are super cushioned with really thick soles come in wider measurements. These are probably the shoes you want to stay away from. This could potentially cause a problem, then, if you have some other injury or ache that requires a thick, padded insole and a lot of cushioning inside the shoe (or if you just prefer to run in this kind of thicker shoe). Shoes that are more minimalist are typically narrower shoes, because they physically have less shoe and are therefore lighter in weight and categorized as minimalist. My personal favorites are Brooks Pure Cadence shoes. They are light enough so that you feel fast when you run in them, but cushioned enough to still feel supported, they last for a long time, and they have a very narrow shape, so they fit snugly against narrow feet. Brooks has other great narrow styles too, such as the Brooks Pure Flows.
Another great option to look into are Mizunos, and particularly, Mizuno WaveRunners. These are going to be a bit wider and have thicker soles and more padding than the Brooks Pure Cadence and Pure Flow Shoes, but the cushioning inside them makes them appear wider than they actually feel when worn on the foot. Plus, they have excellent shock absorbing capabilities for high impact running on pavement and hard surfaces, and they are structurally designed to be long lasting and hold up to a variety of weather, climates, and conditions.