The link to the St Paul's Cathedral website -- https://www.stpauls.co.uk/
Checkout their visit us page -- https://www.stpauls.co.uk/visit-us
And what you'll want is to purchase a sightseeing ticket which includes --
As a sightseeing ticketholder, you will have access to the Cathedral Floor (ground floor) and the Crypt. You can also climb the Dome, up to the three Dome Galleries – the internal Whispering Gallery, the Stone Gallery and the Golden Gallery, which provide incredible views over London. Why not explore our map in advance of your visit?
Please note that the Galleries can be accessed by stairs only. There are a total of 257 steps up to the Whispering Gallery, 376 steps up to the Stone Gallery and 528 steps (total) to the Golden Gallery. We recommend that those with pre-existing medical conditions, mobility difficulties or concerns with heights and confined spaces do not try to climb these stairs.
You will be able to pick up a free multimedia guide and you have the opportunity to join our range of free tours. A video fly-through of the Gallery levels and their views is available on the multimedia guides. You will also have access to any temporary exhibitions and installations, which you can see advertised on our What’s On page.
And
The price of admission includes entry to the Cathedral Floor, Crypt and the Dome Galleries. Admission also includes multimedia guides.
To climb the stairs you're wanting access to the Dome Galleries.
Ticket prices -- https://www.stpauls.co.uk/ticket-types-and-prices