I don't think it's disrespectful to stay open, although if you have a staff member of any race who recognizes that day by doing something specific (here many people use it as a "day of service" and do volunteer work that day -- a tradition I love), then I think you should allow them to take leave the same way you would if you were open for Yom Kippur or Good Friday.
Similarly, I don't know your make up policy, but I might consider it appropriate to clarify your make up policy at this point in this year -- can a family who chooses to stay home that day have their child come another day the same week?
FTR: I'm a white parent in a transracially adoptive family, without the civil rights movement my family wouldn't exist so honoring the movement, and Dr. King is big to me, and I still wouldn't be offended.