A naive reading of Article I, Section 9 of the Constitution would seem to suggest that income taxes, which is the same thing as a "captitation" tax, a tax on a person, are not allowed:
No Capitation, or other direct, Tax shall be laid, unless in Proportion to the Census or Enumeration herein before directed to be taken
The XVI Ammendment created an exception to this rule, but only for Congress, not for the several states.
How is it that states can legally impose income taxes?
First, Article I, Section 9 is exclusively a set of limits on the federal government. The Constitution is designed to establish the federal government; except when it explicitly says so, it is understood to not be describing the powers of the states. States had their own constitutions with their own rules. That's why section 10, which does limit the powers of the states, says "no state shall" do various things. The ban on bills of attainder and ex post facto laws is found in both sections, which is a clear indication that the ban in section 9 doesn't restrict the states.
Second, income taxes are not the same thing as capitations. A capitation (also known as a poll tax) is a head count tax. The tax is something like "$20 tax per person." Income taxes are generally excise taxes, which don't have to be apportioned based on the census. The one other kind of tax that's been ruled to be a direct tax is property tax, and the only kind of income tax that has been considered a direct tax is a tax on income derived from property (e.g. rent). That's why the 16th Amendment says Congress can tax income derived from any source -- even if the source results in something the courts consider a direct tax, it doesn't have to be apportioned.
Lastly, even if state income tax did fall under section 9 and was a direct tax, it would probably comply with the requirement of apportionment. Section 9 doesn't say "no direct taxes can be imposed," it says "direct taxes must be proportional to census population." In other words, states must pay a total tax in proportion to their population. A state can't levy taxes on other states, so "in proportion" should mean 100% of the taxes are on the state itself. State income tax is only paid by state residents and by nonresidents who earned income in the state, so 100% of a state income tax is in fact levied on the state itself.
