Background doesn't have to detract from the subject. Far from it.
There's something ridiculously compelling about a perfectly crisp subject on a beautifully blurred background. It's a technique that we all use because it ensures that our focal point is just that: where the eye wants to look. But sometimes, it can pay to decrease that aperture, lose some of the background blur, and bring more of the scene into sharp focus.
The obvious case for using a small aperture is a landscape shot. Sure, there are going to be times when using a shallow depth of field is going to produce a stunning result, but for the majority of photos, you want to be able to soak in everything that you can see.
But what about other types of photography? Portraits, for example? Well, on occasion including some of the background can enhance your photo, rather than detract from the subject. It can provide some context to where the photo was taken and what was happening at the time. If you're enjoying a trip across Australia, you might well want a discernible Uluru in the background when you photograph your family.
Sometimes, the background is integral to the story that your photo is telling. Without the background reading, as it were, your topic of conversation doesn't make very much sense.
You've got plenty of other compositional tools at your disposal to ensure that your subject remains the subject, not just a shallow depth of field. You can use leading lines, the golden triangle, or contrast to draw the viewers' eyes to where you want them to look. Start by thinking about what you want to be the focal point and work from there to build the composition.
The important thing to remember is that the background needs to be adding to the scene and not detracting from the subject. When you know what your subject is and which story you're telling, stop down that aperture!