Club is responsible for everything that happens on its private property ('within its curtilage') and has to pay for policing services that it requests in those places. If it doesn't request them it won't get them - ring 999 would be the answer. At the Lane that would include the car parks, but not say, the roadway of Bramall Lane where opposing fans mingle pre and post match. That's all on the public purse. The judgement was probably fair but had some perverse consequences for clubs like Chesterfield whose ground sits in a big car park on all four sides of the stadium.
Ipswich Town took Suffolk police to court about 10 years ago and won a judgement confirming this, and that now defines the law in this area. IIRC Portman Road (last there in 1990, 2-2 draw?) abuts the highway on all four sides of the ground, so they significantly reduced their costs as a result.