My understanding of the research is that a higher minimum wage can increase costs, but as a lesser proportion than the increase (edit; to the wage). Labor costs are only portion of expenses for any business, and workers making minimum wage only reflect a portion of the workforce. So, there’s not a zero effect, but I believe it’s usually less than fear mongering would suggest.

I haven’t read into this in at least five years, so happy to admit my own incorrectness if someone knows better.