Solskjaer has had a difficult campaign in his first season in charge, with United sitting fifth in the Premier League.
But they have had some good results, beating Manchester City, Leicester, Spurs and Chelsea, and are still in the hunt for a top four finish.
And Beckham insists the Norwegian is doing a good job and noticed some similarities between him and Sir Alex Ferguson.
“I think he’s doing a good job,” Beckham told Sky Sports. “The fact that he’s stepped in and stepped up, and that he continues to be positive about players, he’s got that from the boss [Sir Alex Ferguson].
“He would never criticise a player from his club and Ole has been exactly the same.
“He’s still learning but he will protect his players and protect Manchester United.
“All of the fans will be behind him because he’s a great person and because of what he’s done for Manchester United over the years.”
United have had a tough time since Ferguson left in the summer of 2013 and have not won the Premier League since.
Fierce rivals Liverpool are likely to win the title this season and Beckham, who launched his Inter Miami MLS side on Wednesday, admitted he is keen to see United step it up again in the near future.
He added: ‘I think when you’re a club as big as Manchester United and you’ve had the amount of success that we had, there was always going to be a period where other clubs have some success.
“When that success is being had by Manchester City or Liverpool it’s always going to be talked about. There was always going to be a transition period, especially when Alex Ferguson stepped down.
“You had David Gill working at the club too and the players stopped playing, Giggsy [Ryan Giggs], Scholesy [Paul Scholes] and the Nevs [Gary and Phil Neville], they all stopped playing so there was always going to be a period when they weren’t as successful.
“Is it hard to watch them now? No because I’m a real Manchester United fan and whatever the situation, I love watching them play.
“But let’s hope it doesn’t last for too much longer that we go without trophies because we are one of the biggest clubs in the world and not many clubs have that tradition.”