Please clarify your question. States have different rules for handling K-1s. Also what kind of K-1s do you have—S-Corp, partnership or estate/trust?
In general, federal K-1 information will transfer over to state and you will have to answer some state-specific questions.
You will still have to pay state tax in your home state on income generated in a state with no state tax. Residents pay tax on income earned everywhere, unless exempted by state law.