The USA is more afraid of Communism due to McCarthy doing a much better job of planning, drumming up and articulating his firm anti-communist stances. Also because a lone actor that blows up a large bank in the name of Communism in 1959 claiming the pigs need to be sent to the slaughterhouse.
So because McCarthy chooses not to drink much and he plans his fear mongering speeches better he becomes the GOP candidate to run in 1960 to be President after setting Nixon up to fall into a political trap on the USSR issue he doesn't quite recover from. He ends up besting JFK in a debate that made him out to be a Soviet supported candidate due to McCarthy making statements that JFK has had affairs with Russian women that are KGB spy's mixed in with some of his stances that he claimed were proof that he was trying to help slowly convert the US to communistic values.
McCarthy losses by 1,000 votes, the EC map by New Jersey and claims that the USSR had rigged the election so he files for a recount in 5 states that he lost. He ends up winning New Jersey in the recount audit and wins the national popular vote by a very slim 3,000 votes. So he becomes President after JFK concedes gracefully but with concern.
McCarthy after being sworn in then signs 3 EO's:
1) A travel ban to or from the USSR and china for a year. Also he orders the CIA to very thoroughly vet anyone who has been to the USSR in the last 10 years and to question anyone who has been there more than 5 times in the last 2 years.
2) He orders the FBI to look mostly in left leaning states for "Communist" that might oppose the current values of the Public.
3) He suspends funding for the Pilot Food Stamp Program because it's an evil unconstitutional Communist program. When questioned on why he did it he says "One who doesn't work should not eat. Only a Communist believes in whole sale free stuff, life just isn't meant to work that way."
He then manages to get the following laws passed due to McCarthy managing to create more fear of Communism than in OTL:
1) A law that makes it illegal to say you are or support Communistic values with a min of 30 days in jail. It's classified as a counter spy law. It spends years being contested in the court system if it's unconstitutional.
2) The Fight For America act: A law with the following
A) The Press must always support America's position's on the USSR to ensure they can't use our own media against the public. Due in part thanks to the USSR influencing the election via the press on McCarthy to attack him with "Clearly pro Communistic values". Based in part on one reporter from the New York Times later found to be from Russia that also clearly had Communist beliefs. McCarthy managed to turn one man into 100 in a speech meant to instill fear into the public.
B) A Federal black list of all possible Communist including but not limited to homosexuals, environmental activist leaders and immigrants from the USSR or china. Any job with a security clearance requirement would have to check this black list before giving a person a security clearance. Though on a case by case basis a person on the black list could get a security clearance if the government agency thought it was worth the risk with a override provision to allow this to happen.
C) A new federal agency called the ACBI or Anti-Communist Bureau of Investigation tasked with Investigating possible Communist would be able to act as cops in all 50 states when needed to fight for American freedom and justice.
D) US citizens that came from any country in the USSR bloc can only buy a gun after a government interview to ensure they do not intend to commit acts of terror or malice on behalf of Communism.
E) People who are known Communist or supporters of Communism can't be hired by any Business with huge fines for those who don't check the Government list of Known Communist.
His VP Richard Nixon worked closely with the CIA to overthrow Castro in the bay of pigs that thanks to heavy US air support ends up killing Fidel Castro. By 1962 a US backed Free Republic is in Cuba and so McCarthy avoids the Cuban missile crisis.
In 1962 McCarthy makes a statement that he going to protect the Republic of Vietnam at all cost and that Communism will not be allowed to spread though violent means. He goes to congress to get approval to go to war with North Vietnam and they give him the green light.
He sends in every air craft carrier and battle ship to North Vietnam and bombs it day and night. He gives the order to the air force to "Burn the reds in hell". By 1964 North Vietnam sues for peace after being fire bombed and shelled 24/7 relentlessly while dealing with a total blockade.
McCarthy then runs in 1964 against George Wallace, what are the odds that McCarthy could win?