It’s been a while since I read the book, but I don’t recall there being anything really objectionable; the rhyme that the book was originally named for is mentioned, but that’s about it.
And yeah, Kipling’s going to be a problem if people ignore the context. He’s interesting in the sense that while he was undoubtedly a supporter of the idea of the British Imperium, and its supposed civilizing mission, he was sharply critical of the actual practice, focusing on the overworked, underpaid and underappreciated civil servants, engineers, and soldiers instead of the elites. And for all that, he’s obviously quite sympathetic towards the various Indian peoples, sometimes more so than the British when the latter are being fools.