Readers of the world, unite ... British Library users. Photograph: Graham Turner

As an occasional user of the British Library reading rooms at St Pancras, I have a recurrent fantasy about sitting next to the ghost of Karl Marx, researching an updated edition of Das Kapital. But after my most recent visit, I'm worried that if he really came back today, Marx might find it impossible to find a seat.

In early April the Humanities 1 reading room was packed, not with grizzled professors but with undergraduates. In its first few years after the move from Bloomsbury, the Library appeared empty and under-appreciated. These days it's starting to feel less like an oasis of quiet scholarship and more like Waterloo station.

The big change came in 2004, when chief executive Lynne Brindley relaxed traditional restrictions on who could have a reader pass. The reading rooms now welcome anyone who wants to do research, including undergraduates. Academics have been complaining for some time; the historian Tristram Hunt accused Brindley of "steadily dismantling a world-class cultural institution" in the Guardian last year and was attacked in turn for spouting "elitist nonsense".

I don't feel happy whingeing about the British Library on the basis of my own recent experience, which I would have rated as "excellent" if I had been asked to complete a feedback form. The Russian books I ordered in advance were there waiting for me; I found a seat and worked happily for two days.

But I couldn't help wondering if all my fellow readers really needed to be there. The young man next to me making notes on Shakespeare's plays seemed to have found a comfortable spot for some Easter revision. He wasn't doing research, and hadn't ordered anything from the Library's stock.

The Library's problem seems to be one of success. Roomy, well-lit, quiet and with comfortable leather and oak chairs and desks, its reading rooms are now a destination of choice. But all the reading rooms together have only 1,200 places, and it's clear that one day soon the Library will have to start putting up "House Full" signs. I wouldn't like to be in the shoes of a professor from Tokyo who comes to London for research and gets turned away.

The Library now seems much more aware of the problem than at the time Tristram Hunt made his complaint. A spokeswoman told me it was looking hard for new ways to "manage occupancy" and had asked universities in London to discourage their undergraduates from unnecessary visits to St Pancras. "Not everybody who asks gets a pass," I was told. "It's still a privilege to get one."

Weeding out readers who not borrowing books is impossible, but passes are now being issued for much shorter periods in an attempt to control numbers. In the long run, the hope is that more and more research material will be digitised and available online.

But if the Library has to choose between keeping academics happy and pleasing the Department of Culture, Media and Sport, then it knows on which side its bread is buttered. "We are funded by a Labour government and we can't be seen to be exclusive," I was told. Any failure to meet targets for widening access would jeopardise its position in the next public spending review.

With parties of schoolchildren tramping in and out, the British Library is not like the British Museum quite yet - but it's heading that way. Coach parties are now welcome, and upcoming speakers include Easyjet founder Sir Stelios Haji-Ioannou. I used to enjoy browsing in the bookshop because of its quirky choice of books, even though it inexplicably failed to stock mine.

It has now been redesigned to look more like a branch of WH Smith, and has big piles of crime fiction and Jeremy Clarkson paperbacks. It's easy to sneer about "dumbing down", and I remember thinking five years ago that the spanking new building was very underused. Has the revolution gone too far? I wonder what Marx would think.