How did the Beatles influence politics?

They generally stayed out of politics; Ringo Starr's approach to war was "Put the politicians in a boxing ring, and let them battle it out." British PM Harold Wilson recommended them for MBE medals, in an effort to gain popularity himself; it worked for awhile. When he met Lyndon Johnson, Johnson said "I like your advance guard (the Beatles, who had recently toured the US), but don't you think they need haircuts?"

At the height of Beatlemania, the Duke of Edinburgh commented "I think the Beatles are on the wane." This was quickly softened to "I think the Beatles are away" in the press, when public opinion threatened to turn against him. Occasional statements were also made by Buckingham Palace, about the band and their doings.

John Lennon returned his MBE medal in 1969; the gesture didn't help him later, when he asked for a Royal Pardon for his 1968 hashish conviction, and neither did his criticism of the British government. Richard Nixon was scared of his potential influence, and had the FBI tail him for months, after he moved to the United States. Senator Strom Thurmond recommended that someone find an excuse to deport Lennon, and he spent the next four years battling the INS. After his murder, both Jimmy Carter and Ronald Reagan publicly expressed their condolences. (Reagan and Lennon met once, guesting together on "Monday Night Football".)

George Harrison hated paying taxes, and didn't care who knew it; his song "Taxman" wasn't written for nothing. (All the same, when the Inland Revenue told him he owed them another £3 million in 1973, he wrote them out a check on the spot, just to get it over with.) He visited the White House in 1976 and met Gerald Ford. He also rubbed elbows with Prince Charles, at the Prince's Trust concerts; it was Charles who suggested Harrison form the Travelling Wilburys, and the name came from him.

Paul McCartney was knighted for services to Queen and country; mostly for his charity work in Britain. He never spoke openly against the government, and usually gave his share of lip service to them.

Ringo Starr moved to Monaco, as a tax exile from Britain; what he was expected to pay in taxes was crippling him financially.