How did the U.S. affect East Asia in the 1800’s and 1900’s?

The US has had very different effects on East Asia at various points in the two hundred years that this question spans.  These effects have been felt through both trade and war.

The US started to impact East Asia in the mid-1800s.  At the time of the first Opium War, in the 1840s, the US (with British help) forced China to grant it special privileges within China.  The US was to have trade privileges in China and its citizens were exempt from Chinese laws.  Not long afterwards, the US had a major impact on the other main power in East Asia, Japan.  In 1853, American warships forced Japan to open itself to foreign trade.  For the rest of the century, the US mainly impacted East Asia through its trade.  Trade exposed East Asians to ideas from the US and Europe, changing the cultures of East Asian countries to some degree.

American trade with East Asia continued to affect the region throughout much of the 20th century.  This is particularly true of Japan, which was heavily influenced by American culture in the pre-WWII 1900s.  Japanese started to wear Western clothes and enjoy American music.  They started to play baseball.  They watched American movies.  Many English words entered the Japanese language.  This trend towards Westernization broke for a while during WWII, but then continued through much of the 20th century.

The US also affected East Asia militarily, mostly during WWII and the Cold War.  The US, of course, devastated Japan with bombing raids towards the end of WWII.  It then used its military might to impose a new constitution on Japan and to occupy the country for a few years after the war ended.  The US impacted Korea by dividing the country with the Soviet Union and then by fighting a war there.

After WWII, the US has mainly impacted East Asia as a trading partner.  US trade helped both Japan and South Korea boom in the post-war era.  At the very end of the 20th century, trade was starting to help China rise to greater wealth.  As the US has traded with these countries, it has created a complex relationship with them where the US impacts the Asian countries culturally and intellectually while running trade deficits with them and boosting their economies.

In these ways and many others, the US deeply affected East Asia during the two centuries that you mention here.