Presidents used to a large open reception on New Year's Day - anyone could stand in line to shake the President's hand. George Washington started this tradition and John Adams continued it, holding the first one in the White House itself:
The first occupant of the White House, John Adams, took up residence in the unfinished mansion in November 1800, and hosted its first New Year’s reception on January 1, 1801.
A history of the White House published a century ago noted that John and Abigail Adams hosted a “very formal affair”:
The President and his wife did the honors alone that New Year's Day, and it does not seem to have occurred to them to call on the Cabinet families to assist them. The President's wife sat in state in her brocades and velvets, while the President stood beside her in knee-breeches, gaily colored waistcoat, high stock collar, and his powdered hair tied in a neat queue. After each guest had paid his respects to them, he passed on and was served with refreshments by a waiter.
The Madisons even continued it after the White House was burned:
Following the burning of the White House by British troops in 1814, the New Year's Day levees were held in the rented houses used by presidents James Madison and James Monroe.
The White House receptions resumed on January 1, 1818, hosted by Monroe in the rebuilt mansion. At that time it was decided to hold an earlier reception for the foreign diplomats and government officials, so they wouldn't be subjected to the crush of people in the public reception.
Customarily, anyone waiting on line outside would be admitted. After greeting the president in the Blue Room, the crowd would be directed into the enormous East Room. A temporary wooden bridge would be positioned in one of the large front windows of the East Room, and the guests would exit through the window onto the White House lawn.
Lincoln shook thousands of hands on January 1, 1863 before signing the Emanicipation Proclamation. You can read a Time article from 1930 about a man who was first in line in 1930 to shake the President's hand:
Long before New Year's daylight John W. Hunefeld arose in his Washington home on C street, put on his salt-and-pepper suit, breakfasted quickly, and set forth through dark streets for the White House. He would do no odd jobs of painting and-carpentering that day. He was going to meet the President.
Citizen Hunefeld had reason for his early morning haste. Twice before had he proudly led the public line into the White House for the New Year's Day Reception (a custom originated by President John Adams in 1801) and now he almost ran to be again the first at the White House gate on Pennsylvania Avenue.
He got there on time. After sunrise a woman in a brown fur coat, a Mrs. George A. Barrett, lined up behind him. With satisfaction Citizen Hunefeld surveyed the lengthening queue stretching out along the fence, down the street, around the corner. Fine weather, good spirits and natural curiosity about a new President had brought out thousands of plain people who do not get written invitations to White House functions.
Before noon officials began to arrive in their motors to greet President & Mrs. Hoover. Citizen Hunefeld recognized many of them—Vice President Curtis, Speaker Longworth, the British Ambassador in his red jacket. Finally the great White House gates swung open and Citizen Hunefeld marched grandly at the head of his procession up the curving sidewalk to the big glass doors of the White House. When these opened Citizen Hunefeld did a gallant thing: removing his cap, he stepped aside to allow Mrs. Barrett to lead the line past the President.
With the Marine Band dinning in his ears, Citizen Hunefeld took note of the bodyguardsmen (secret service) standing about. They could not be too careful guarding the President's life. Some crank might get in. McKinley had been shot that way by a man with a revolver under a handkerchief. President Harding had been asked to wear a bullet-proof vest at his first reception in 1922 but refused. An experienced receptionist, Citizen Hunefeld knew he could not put his hands in his pockets; he had seen women warned to take their hands out from under their furs.
In the Blue Room stood the President. Beside him was Mrs. Hoover in a blue silk dress. Close at hand was Col. Edward Starling, Chief Bodyguard, looking not a bit like a detective in his cutaway.
"A Happy New Year, Mr. President!"
"A Happy New Year to you." And the President gave Citizen Hunefeld's hand a hearty shake. Like many another, Citizen Hunefeld observed that President Hoover's handshakes were less rushed than President Coolidge's, that the reception line moved along more leisurely. In all that day President Hoover greeted 6,348 officials and citizens, the largest New Year's reception in many a year. The warm air in the White House, the heavy scent of flowers, perfume and outdoor clothing, drove the President out on the rear portico for fresh air twice during the three-hour ceremony. As he returned the first time, he said to Mrs. Hoover: "First down!''
The last reception was held in 1932 by Herbert Hoover, who broke the tradition the next year and it was never revived.