The Star Trek series has many different classes of star ships featured in the shows and movies. Some examples include: Constitution-class, Galaxy-class, Excelsior-class, Intrepid-class, Defiant-class and Sovereign-class. But never have I seen or heard reference to any Enterprise-class ships.
Starfleet ship classes are usually named after the first prototype vessel of that class made; i.e. the first Galaxy-class vessel would have been named the USS Galaxy. And as prototype vessels usually have NX- designations that leads me to believe that the class of ships seen in the show Enterprise would be of an Enterprise-class, if the design was ever put into production. But as far I am aware that never happened in canon and the NX-01 Enterprise was referred to as a NX-class ship on the show.
To sum up then, has Starfleet ever had an Enterprise class of star ship?