"Harrison Bergeron" was published in 1961, but many of the concerns Vonnegut was addressing at that time are still present today.
Most prominently, Vonnegut is satirizing the discriminatory attitudes of a society that says it is bringing all people to equality. In the story, set in 2081 and governed by the "211th, 212th, and 213th Amendments to the Constitution" as enforced by the Handicapper General and her agents, differences between people are masked by handicapping devices. In today's society, many if not most people would claim that all individuals are treated equally - unless a person has too many tattoos or piercings, or the hair is too long, or the name and appearance are too Arabic in origin, or ...
Vonnegut is also satirizing the impact of the mass media. George and Hazel seem to get all their information about their world from their television, although they don't remember any of it for very long. Vonnegut would probably not be pleased with the proliferation of news channels on television and websites on the internet, each offering one interpretation of events that many people accept as completely accurate without further research or validation.