Developing callouses will help. If you have to press too hard your fingertips will expand on the fretboard, touching other strings.

Correct hand position will help, make sure you're coming straight down on the strings.

Also make sure that your elbow isn't "flying out in space" or "chicken winging". I had a problem with this for a long time when I was beginning. Fingertips are not symetrically round. They tend to be wider from side to side. So if you turn your fingers away from being parallel to the strings there is a good chance of touching the string above or below. Keeping your elbow close to your body will help your fingertips align properly with the strings.