No, you do not have to amend your return over a correction in the employer's state ID number. That information has no effect on your tax calculation.
The state employer ID number is really only a bureaucratic convenience for your state's department of revenue; it's simply not an essential element to the correct filing and processing of a valid personal income tax return (either state or federal).
(if, however, the state advises you that they couldn't verify state tax withheld, you may have to amend. This is unlikely).
