What do I need to do to have state taxes taken out of my check if none is being taken out?

Why aren't they taken out? If you have to pay State taxes your employer is required to deduct them - of course, how much depends on total amount of income, dependents, etc.

Talk to your payroll department, if you earnings are low and you have a large amount of dependents often there will be little or no state or federal taxes taken out.