There is a Grayson Perry etching on the wall in Andy Burnham's office. It is called Print for a Politician and depicts all the types of people in society. There are Elitists, Townies, Fat People, Parents, Methodists, Smokers and Romantics. “I'm over here - I'm in the group called Labour and I'm one of the Oiks,” the Culture Secretary said. “We are too quick to categorise people.”

Mr Burnham does not fit easily into any box. He is a Liverpool lad who read English at Cambridge. He drinks his tea from an Everton mug, but he can also recite Shakespeare and Keats. He is, at 38, one of the Cabinet's “Young Turks” and he is an old-fashioned Roman Catholic boy who believes in marriage and morals. He is a Blairite who switched to Brown.

He and James Purnell, his predecessor at the Department for Culture, Media and Sport, are the rising stars who are now rivals as well as friends. “I'm purging the Purnellisms,” he joked about his new office.

After this week's by-election in Glasgow East, the new generation of Labour ministers must be wondering what the future holds. “We have taken some hard knocks but we must pull together,” Mr Burnham said. “Even in difficult times there are things to feel optimistic about.”

The Minister for Fun is refusing to mope. “I have a fantastic job,” he said. “I have been to the opera and Wimbledon, it was the Proms this week and I love Tate Britain. But what I really enjoy is going to sport. I would not normally have gone to things like gymnastics and cycling.”

He thinks his job as Culture Secretary is to cheer the rest of us up too. His proudest achievement has been to introduce free swimming for all. It was he who as Chris Smith's special adviser helped to push through free museums and galleries and he is currently working on another generous plan. “You could do the same for the performing arts,” he said. “Free theatre and opera weeks, I think it's a brilliant idea.”

In his previous role as Chief Secretary to the Treasury, Mr Burnham was responsible for putting the brakes on spending for the Olympics. He still insists that there must be “real rigour” in the analysis of the 2012 accounts but he thinks the money will be well spent if it creates a new generation of winners. He wants Britain to come in the top ten in Beijing and he said: “In London we want to go farther still.”

Although he admitted that the cost of the Games “is something people will keep coming back to” - particularly during an economic downturn - he said: “The benefits will be huge. This is not just some big two-week party. We won't regret it.” His sister-in-law has been lobbying him to add netball to the list of Olympic sports. His own favourite is cricket - “It's a myth that all I do is football,” he said.

It is, however, the beautiful game that he wants to change. Mr Burnham believes that there is a serious problem with the number of foreigners playing in the Premier League. “I'm not xenophobic in any way but I care about the health of English football, the state of the grassroots game, the quality of the competition and the ability to win of the national team,” he said.

“Most football supporters want to see their clubs have the best in the world but they also want to see young local talent being brought on.” Football's international governing body, Fifa, has proposed that football clubs should be forced to play at least six home-grown players in their teams. Mr Burnham would not be drawn on what the number of English players should be but he supports the principle of a quota in order to boost football in this country.

“There is an argument both in terms of supporting the national team and helping secure more equal competition within the Premier League for a system of quotas.”

England would in his view have a better chance of winning international competitions if it encouraged its own players. He was horrified that the national team did not even qualify for the recent Euro 2008 championship.

“It was a bitter pill to take not being at Euro 2008. We have to ask some hard questions about whether we are doing enough to give young English players a chance. We need to maximise the home-grown talent for the national team to draw on.” There must, he thinks, be national rules because without them the clubs would all compete to recruit top players from all over the world.

“It is like an arms race, bringing in the best from outside. It would be more healthy to get more home-grown talent and encourage clubs to train and nurture people.”

The plan could fall foul of European law on the free movement of workers, but Mr Burnham is ready for a fight with Brussels in what he believes is a populist cause. If the football authorities can agree a blueprint, he said: “I would be quite prepared to go to Europe and say ‘this is our proposal let's see if we can get clearance for this'. The debate is around applying EU law sensitively.”

The Culture Secretary doesn't want a cap on what players are paid - “I am not going to dictate people's salaries,” he said. But he thinks football club owners should be more responsible with their resources. Billionaires such as Roman Abramovich should, he says, follow the lead of Randy Lerner, the chairman of Aston Villa, who has given £5 million to the National Portrait Gallery. “What matters is not someone's nationality but their intentions when it comes to owners,” he said. “Most people think, ‘Have they got the best interests of my club at heart?' In the low divisions people are asset stripping, that is what football should stop.”

There has been discussion of stripping of another sort in the courts recently. Many in the media are furious that Max Mosley, the chief of world motor sport, won his case against the News of the World, which accused him of taking part in a Nazi-inspired sadomasochistic orgy. Mr Burnham, however, thinks that the case shows that the law is working. “The accusation is that the law has always been too weak,” he said. “But it now appears that under the current law people can protect their private rights.” He does not think a privacy law has been introduced by the back door. “For people in public life the test should be whether their private activ- ity impacts on their fitness to do the job. That should be a test done on a case-by-case basis.”

The Culture Secretary also backed a revolution in the way in which the internet is policed this week. Parents will in future be punished if their children download music illegally and people will be expected to pay for the tunes they get online. He admits that his son has to load up his iPod with music every Sunday - “I don't know how to do it but I always pay”.

He believes that the law had to catch up with a world that has changed dramatically since he grew up watching The Magic Roundabout and Mr Benn. “If you ask children now to choose between a television and a computer they would choose a computer,” he said.

“When the internet arrived some powerful ideas came with it, such as content is free, but that wipes out the value of the creative industries and does the country a major disservice. Most people are prepared to pay a reasonable amount for their music.”

The BBC may have to pay a price for the new digital era. Mr Burnham is considering giving some of the license fee money to other broadcasters. “We have the best broadcasting system in the world and the question is how do we sustain it in a very different era.”

The Labour Party is also fighting for survival. The Glasgow East by-election is only the latest blow. Mr Burnham doesn't think a new leader is the answer. “The party needs to pull together and pull behind Gordon,” he said. “We've done a lot but we've been in power a long time and people say - ‘where are the ideas?' The onus is on people like James [Purnell] and me to come out more and articulate our ideas.”

Labour, he says, must rediscover a sense of optimism. “The Government should be talking quality of life and general wellbeing,” he said. “We should emphasise education in the broadest sense rather than just numeracy and literacy. People want happiness rather than just money.”

The next few months will be critical for Labour as well as for Mr Brown. “The question for me,” the Culture Secretary said, “is not, ‘Has Gordon got the character?' It's, ‘Has the party got the character to keep its nerve to pull together in difficult times?'”

Times Recommends

• Everyone will get vaccine against swine flu
• Artificial sperm and the race to build a baby
• BBC staff lavished with trips to lobby for awards