The first-round demolition in front of an appreciative home public at the event where the 17-time Grand Slam winner got his start in the game as a ballboy took only 54 minutes.
The 2014 finalist David Goffin of Belgium, hammered by Federer in less than an hour a year ago in the title match, won his opening match as he beat Italian Andreas Seppi 3-6, 6-4, 6-4.
The fourth-ranked Federer was rock solid in his first match since an opening loss a fortnight ago in Shanghai.
The Swiss is seeking his tenth straight finals appearance at Basel.
He last beat his 64th-ranked Kazakh opponent in the Davis Cup quarterfinals in Geneva a year ago.
Federer said he knew that he would be facing a tough customer in Kukushkin.
"I was seeing the ball as big as a melon," said the 34-year-old, who has five titles this season.
"I knew that I had to stay with him in the first set as he would be looking for a fast start," the Basel native said.
"That was my main strategy, to keep him under control," he said.
Federer said he knew the second set would go easier.
"I'm very happy with this performance, especially to advance so well here at home," he said.
"He was trying to give me a game I don't like but I was able to counter that.
"This was a very good start for me, an enjoyable match."
Grigor Dimitrov began erasing a run of recent poor form as the Bulgarian beat Sergiy Stakhovsky 6-3, 6-4 to reach the second round.
German Philipp Kohlschreiber, meantime, put out Pole Jerzy Janowicz 6-4, 3-6, 6-3.