Nothing at state level, and that is the whole story
The South Dakota Department of Revenue puts it plainly: the state is one of seven that does not impose a state income tax. There is no withholding certificate to file, no state allowances to claim, and no paid-leave or disability premium to deduct. The only thing your employer sends the state on your behalf is unemployment insurance, and that is the employer's own cost rather than a deduction from your pay.
Where the money comes from instead
Sales and excise taxes fund the state, which is a different bargain rather than a free lunch: nothing leaves the paycheck, more is taken at the till. For anyone weighing an offer across the border, the take-home difference against a taxed neighbour is exactly the state line you will never see on the stub.