Last time I was in the British Museum (about 15 years ago I think) there was just a huge glass box in the foyer where you could put a donation. I don't know how much of the costs it covers but I saw quite a few people put substantial amounts of money in there. I paid five pounds and had a rather pricey snack in the café as well. No queue to get in.

It also works because you would feel like shooplifting if you pay nothing, specially if its your first time in it.