State income taxes are a means of funding services within the state but they are also, like federal income taxes, a means of redistributing wealth. The benefits of having a state income tax therefore depend a great deal on what your income is. Wealthy people will generally be against state income taxes (assuming that they are making the decision in their own interest), while poor people will be in favor. One might also state this point in terms of political attitudes to fiscal policy. Conservatives will usually be against state income taxes, whereas progressives will generally be in favor.
There are a few refinements to this general position. States with no personal income taxes will either have to provide fewer services, or make up the difference in property and sales taxes, or (most likely) a combination of the two. Those who would pay high state income taxes will also be likely to pay high property taxes, though there are exceptions to this rule. Similarly, people with high incomes spend more money, and thus pay more in sales tax, though this will seldom outweigh the income tax saving. Wealthy people are more likely to be able to arrange their purchases, particularly of high-value items, in such a way as to avoid state taxes.
Depending on which services are unavailable because there is no state income tax, some wealthy people might decide that it is worth paying more in tax to secure these services. This is particularly likely if these are services which are difficult or impossible to purchase privately, such as high-quality infrastructure, or the maintenance of law and order.