As **Canadianer** already mentioned in the comments, the majority of bacteria (this includes gram-negative bacteria) are non-pathogenic.
I think a prime example is the human gut flora. You can find loads of bacterial species living in your gut without doing you any harm, if anything they are doing you a favour by breaking down food.
Here some examples of gram-negative genera:
* _Bacteroides_
* _Escherichia_
Most species within those genera are non-pathogenic, however, there are exceptions.