American influence on Japanese and Korean language has been pretty strong. Both Japan and Korea have been home to large numbers of US troops since 1945 and 1950, respectively, both countries have had close economic ties with the US, and both countries are very interested in popular culture.

Here is a video of Japanese greasers dancing in Yoyogi park. Yoyogi park is home to the Meiji Shrine, the shrine for the first emperor of the Meiji restoration, the guy who started Japan's industrialization and subsequent militarization. It's a an odd scene, something so American in such a Japanese place, but it's certainly not unique to this place alone. The Tokyo Dome hosts the Rice Bowl every year, an American football game between the best college team and the best corporate team. It's been played since 1984 and the X-League has been active since 1971. Japan is also big into baseball with numerous players competing in the US MLB.

Japan has frequently taken American words and adopted them into English, including baseball and American football which have become besu-boru and Amefuto, respectively.

Korean has done the same thing but not necessarily with the same words. In Korean, baseball is called 야구 (yagu) and it's rare for people to refer to it as baseball. The Japanese word for baseball is 野球, pronounced as yakyu but it is frequently referred to by it's Japanized English name, besu-boru.

As to your question, how much influence has the US had on the culture and language in general, it really depends on the topic. Modern words are more likely to be loan words - computer, television, radio, smart phone - than older words. Even though the words are adopted doesn't mean that they will be the same. For example, in Japanese and Korean they use the word "apato/apateu" from "apartment" and use it to refer to their homes. What they refer to as apartments we would generally refer to as a condominium - a unit in a building that is independently owned, not rented. Also, they will frequently mash two words together, so a word like "remote control" becomes "remocon" in both languages.

It's interesting to note that while South Korea willingly adopts English loanwords, North Korea will not. Here is a video of North Korean and South Korean words and how different they are. A few notes - the pepper translates as "lion pepper" in North Korean and the South Korean girl refers to it as 피망 (pimang). The more common way to refer to a bell pepper is as a 파프리카 - paprika. During the book exercise, the first word the North Korean highlights is "digital" and the second is "internet." It should be noted that the words immediately preceding internet were "radio, newspaper." Based on context, anyone in a modern country would probably be able to know that "internet" goes with it. In North Korea, rather than adopting a loan word, they create "pure Korean" words instead. This generally means that they will not use words of Chinese origin, but only pure Korean. One issue with this is that pure Korean (like pure English) words are so old and for very basic ideas that it can be difficult to adequately describe newer ideas or inventions without sounding incredibly strange.

Finally, it should also be noted that Japanese and Korean both have adopted words from other languages - arreubait is from German and it means part-time job, piman/pimang (bell pepper) is from French, and the word "gipseu" is used for a plaster cast, like for a broken arm. It comes from the word "gypsum" but it's of German origin as well. Personally, my students have looked at me like I was an idiot when I didn't know the word gips. "But teacher, it's English!" Cue my eye roll...

TLDR - Japanese and Korean have both been influenced by English words, as well as many other European words for mostly newer, modern concepts or things. The words don't always have the same meaning when used however and the languages are full of loan words from other languages as well.