When I was growing up 50 plus years ago, it was pretty much a given that dogs would chase cats -- it was almost like a chemical reaction, owners couldn't control the dog and you simply had to keep them apart.
It seems to me based on videos, etc. that many dogs now will do things like "adopt" kittens and in general no longer chase them automatically. Now, I am sure that I am seeing a non-random sampling of behavior, since videos of dogs simply chasing cats would not be very interesting. Moreover, the Internet has served to increase awareness of unusual and/or cute/intelligent behavior of dogs -- there are literally thousands of times the amount of footage of pets since Youtube, etc.
However, I do think it is possible that the past 50 or 60 years could have made a big difference in behavior in both dogs and cats. If we compare the pets to humans in terms of breeding, there are two major factors: firstly, the much shorter generations (I think a cat can have kittens as young as a year old or younger) and secondly, pets are in fact bred whereas humans are not.
And if indeed humans have impacted dog and cat behavior through selective breeding, I am sure that both species are kept in general for very different reasons today. 150 years ago, given that many humans struggled to get enough food themselves, both dogs and cats were more or less working animals -- dogs for protecting livestock and property and cats I guess mainly for mousing. It was much more common for dogs especially to remain outside their entire lives and farm cats also would have lived in barns but not inside their owner's house. Of course, some dogs and cats were kept as pets but it was probably fairly rare.
Once humans started doing better in terms of survival, I would guess animals kept purely as pets would have gradually become a more common thing but perhaps it was post WW2 when many Americans became suddenly prosperous and home ownership became more common that keeping pets and not work animals really would have taken off -- I guess that is a separate question, whether WW2 did indeed cause an increase in pet ownership.