Howard Hawks said that this was, technically, one of the hardest movies he ever made because they were "working with leopards and dogs." Except for a few scenes early in the schedule, the stars couldn't usually get too close to the leopard, so all kinds of processes were used to make it look like they were interacting with "Baby." Apart from the usual practice of using a body double, they used glass partitions, split-screen, rear-projection, and so on.
RKO, which is my favorite of the Golden Age Hollywood studios (they seemed to make more varied and unique films than any other studio), probably had the best special-effects department in all of Hollywood, led by Vernon Walker. Three years later, of course, his department did its most famous work on Citizen Kane, where special processes were needed in virtually every scene. The job on Baby is just as impressive; I didn't really become aware of the rear-projection until long after I had learned to spot rear-projection in every other Hollywood studio movie. In the scene where they're in the car with Baby, it's not hard to see that the road is rear-projected, but I didn't guess that Baby was part of the screen rather than part of the set.
Update: As noted in comments, the optical printer used to achieve some of these effects was created by Linwood Dunn; this is also mentioned in the description for the video itself:
This is patterned after a talk I saw by Linwood Dunn, who worked with Vernon Walker on the many special effects needed to keep the leopard separate from the actors.