This is a bit of a loaded question as it presupposes that Americans today are generally anti-union. I'm not sure if that's 100% true, but I'll go into detail about why people might have a negative perception of unions in the USA.
Not a historian, but fresh off a Labor History course pertaining to the United States; delete if that doesn't fit your qualifications but I'll cite the textbook I'm using/used.
Public support of unions tanked during the mid- to late 70's as a result of a few different factors, culminating in what we retrospectively could view as a "final blow" from President Reagan in 1981 (we're all at least somewhat familiar with the PATCO Strike.) But let's start from the beginning.
Unions gained varying degrees of success organizing miners, steel and iron workers, and various other forms of laborers through the end of the 19th and beginning of the 20th century. Union strength continued on an upward trend as the American Federation of Labor (AFL) was increasingly popular as a "craft" union (craft refers to unions organized by people's "craft" or their trade; if you were an auto worker who made tires, you unionized only w/ fellow tire makers). The Congress of Industrial Organizations (CIO) arose in 1935 as a split from the AFL to contrast their craft unionism and employed industrial unionizing (if you worked in the same industry, you unionized altogether; tire makers, engine makers, assembly workers, etc). The CIO tended to attract more socialists, communists, anarchists, and leftists in general as their form of unionizing was seen as more all-encompassing, and better for workers at large, even if all the members of the union weren't on the same skill level. Ask r/socialism101, r/communism101, and/or r/anarchy101 if you want to know why they all supported/still support mass unionization!
The first major hit to union strength was the Taft-Hartley Act of 1947, combined with the McCarthyist practices of the 1950's. The Taft-Hartley Act removed some protections of labor unions, making things like wildcat strikes (strikes w/o approval of union leadership), picketing, secondary boycotts (strikes or boycotts in support of another union), and federal employee strikes illegal under provision of the prior National Labor Relations Act of 1935. It also disallowed communists from being in unions; this part doesn't mean much in practice, as one could simply lie, but partnered with the Second Red Scare that followed 1947 (emboldened by McCarthy himself), many labor union activists were subject to investigations, aggressive questioning, and even detainment. While these investigations rarely led to anything damnable in the eyes of the law (i.e. actual Soviet spying), they resulted in the loss of many union jobs and loss of union leadership amongst the more left-leaning CIO, which was eventually reabsorbed into the AFL, forming the AFL-CIO.
The 60's and early 70's saw moves towards more union strength, with the new service industry seeing some unionization. In addition, women's unions gained strength fighting degradation, including the unionization of stewardesses (what we would call flight attendants now). A major signifier was the wildcat strike of postal workers in 1970; a strike without union approval that spread across the country. President Nixon ended up calling in the National Guard to deliver mail, but the strike lasted only roughly 8 days before he signed a new law forming the United States Postal Service and granted them collective bargaining rights, wage increases, and other benefits. Around the mid-70's, however, is where public opinion took the biggest hit.
The 1973 Oil Crisis (among other issues; I'm not an economist!) resulted in a major recession lasting a few years. During this time, anti-union sentiment was stoked hard due to union members being paid more than many non-union members in similar or the same industries; blue collar workers tended to blame those that were unionized for their unemployment or their low wages. This sentiment is partially what helped elect President Reagan, a conservative republican who spoke to middle class values. This proved to be what many consider the death blow to labor and union strength.
In 1981, the Professional Air Traffic Controllers Organization (PATCO) went on strike for better working conditions, higher wages, and a shorter work week. This was by all accounts an illegal strike, as they were federal employees and did not have the right to strike. Reagan actually had the support from PATCO for his presidency, as they were primarily middle class, blue collar workers that he proved very popular amongst. The president immediately ordered the 13,000 air traffic controllers back to work, and only 10% of the returned. The next morning, he gave them 48 hours to return to work, citing the aforementioned Taft-Hartley Act that prohibited government employees from striking (remember that one?). Over 11,000 air traffic controllers were fired, which was seen as a HUGE display of power, strength, and resolve from the new President, increasing his popularity and really tarnishing the reputation and cementing the negative general opinion of unions among many people, particularly baby boomers and older who can remember the recession of the 70's and Reagan's presidency. As an aside, union membership is currently at roughly 10-11% depending on how recent your data is. In 1983, it was at roughly 20%. Peak private sector unionization was at 34% in the 40's, and it's now at 7%.
Some of the background information of this may seem only tangential, but I felt a deeper understanding of the history of unionization and labor in the united states was necessary to understand why public perception of contemporary unions can be negative!
Sources:
Dubofsky, Melvyn, and Foster Rhea Dulles. Labor in America: A History. 8th ed., Wiley-Blackwell, 2010.
U.S. Bureau of Labor Statistics (bls.gov)
Hi OP and potential respondents. We've allowed the question, but contingent on the understanding that this question is about history, not present-day conditions. When answering questions here, be mindful of the "no current events" (aka "20-year") rule and confine answers to pre-1998 only. Thanks!
Not to discourage other responses, but you could try asking this same question in r/asksocialscience as well.