Static methods aren't polymorphic, so you are missing one of the main advantages of OO if you use them - they inherently lead to a procedural design.
Static classes, as the latter approach often is called, are mostly useful as utility classes, such as java.lang.Math. They should mostly be avoided, in my opinion.
The soul is dyed the color of its thoughts. Think only on those things that are in line with your principles and can bear the light of day. The content of your character is your choice. Day by day, what you do is who you become. Your integrity is your destiny - it is the light that guides your way. - Heraclitus