I'm not sure exactly what you have, so I'm going to be very basic here. My apologies if it sounds patronizing, that is not my intent.
First, I'll make sure you have horses as shown here in the very top-left:
I have 192 horses available. If you have more than a dozen or so horses available then things look good. If you have 0 horses there's some problem (something not researched, your horse resources aren't built or are still building, etc.). If you have only a few horses you may need more than that for the unit (but the unit should still be in the list of buildable units...).
Next, try to design a mounted unit. Pick a city, select "Trai"n then "Design" then "New" then "Equipment" and select horse as seen here (horses are under the "mount" bit, I tried to post a bigger screenshot which included the horses but the forum didn't like it so had to reduce the pic to this size...):
Give the new unit a horse and nothing else -- no weapons/armor/traits/etc. -- then "save" and name the unit ("temp horse" or the like) then close out Design. By not giving the new unit any weapons/traits/armor/etc you don't have to worry about having crystal/metal/etc., just horses. The new unit should have a horse in its picture. Go to your city then "Train" and see if the new unit "temp horse" is available to train. If it is then everything is working fine and you must not have a default horse unit available (I tend to Hide all default units and create my own so I don't know if/when default horse units appear).
If "new horses" isn't available, then there's a problem. Let us know that and we'll see what's what. I'm hoping this isn't the case as I'm not sure what the problem would be...