There is a correct answer here. To say it this way, as has been discussed, is not entirely natural, but I'll get to that. Saying it like this puts emphasis on the fact that you are interested in how they walk. To say: "I am sitting in a cafe watching people walk" is grammatically correct. "Walk" is the verb for "people". To say "I am sitting in a cafe watching people walking" is not grammatically correct. "Walking" is a gerund and shouldn't be used in this sense. You could say: "I am sitting in a cafe watching people who are walking" if you wanted to. However, to get your meaning across, you really need an English colloquial phrase: "people watching". It literally means to sit somewhere simply watching humanity go by, implying also a relaxed attitude and satisfaction with your circumstances. So, to get your idea across I would say: "I am sitting in a cafe people watching." If you want to focus on that aspect that the people are moving by, you could still say: "I am sitting in a cafe watching people walk by." Hope this helps.

"I'm sitting in a cafe watching people walk or walking?" For me, both don't sound really natural. If I'm in a cafe and I'm watching people walk, I think I would be examining the way they walk - fast, slow, with a limp, etc. and the same would be true for "walking". Normally, if I'm sitting in a cafe, I'd say that I was watching people walk by (or "passing by") since I would only see the people for a specific amount of time and then each of them are gone. The same could be say for other locations where you are stationary and are observing motion: - I'm sitting on a park bench watching people walk around the lake (across the grass, along the river, etc.) - I'm standing in a concert hall looking at the crowd flowing in - I'm lying on the grass watching the clouds parade across the sky

From a native speaker, "I'm sitting in a cafe watching people walk" sounds more natural. If you want to try and create a difference between the two, then "watching people walking" makes it sound like you're super focused on the act of walking and less so on the people. But people will know what you mean either way.