Simply put, the most important function of rain gutters is to protect your home from the harmful effects of water.
If you have gutters that are old, damaged, or missing parts then you are risking damage to your home. It can prevent the gutter from doing what it is meant to do which is to keep water away from the edge of your home - and a wet and saturated ground can produce a whole host of problems.
If the ground around the foundation of your home is too wet, then it can be a breeding ground for mold and mildew. This can eventually lead to structural problems which in turn can cause cracking and leaking, especially in the cold season when the moist ground can freeze and in turn expand. This can lead to serious damage to the structure of a home - and something that can be easily be prevented with the presence of a quality gutter system. A properly installed gutter will manage water runoff properly and guide it down through the downspouts to appropriate areas.
The basic principle here is that an "ounce of prevention" can be a worth a "ton of cure" when it comes to properly managing water runoff around the house.