Internet should be capitalised, because it's a proper noun and defines a single, definite thing not something general (like the word tree, for example).
If you look up the word, you'll find it always capitalised, I can link you to the OALD, as an example.
If you're writing in some informal context, most people won't mind if you write it lowercase, though. But if we consider some official context, for example writing your Curriculum Vitae, you would write it properly.
EDIT: After Garet's comment, I decided to introduce a distinction. If you want to refer to a general inter-network (which by the way, it's the compound where "Internet" comes from) or if you want to use the same term to refer to something which is not the World Wide Internet, then it would be written lowercase.