The same way they are today - led down a ramp into a stall area (3' x 10" +/-) and tied. They generaly made the trip standing up without exercise or ability to walk around. Fed hay & water during the trip. Did their best to stay standing in rough waters. Depending on the era and the cost - some were treated better than others. Mortality rate for Spanish crossings of the Atlantic (think Cortez, etc.) was 50% +. In the days of Noah trips were shorter.
Today you can ship your horse by Fed Ex - loaded into one of those baggage containers and fork lifted into the plane.