Do American universities generally sponsor H1B visas for staff?
Yes.  A simple google search for "H1-B university staff" links to a bunch of federal and school-specific policies.  It goes without saying these roles are reserved for people with "specialized knowledge," etc.  Also a link to a similarish question about faculty is here.

You asked "what type" and I missed that the first time around.  Here is a link to a page that explains H1B jobs open at NYU.  Not all but mostly faculty positions.