In reply to Godwin:
Some routes into climbing/ mountaineering are definitely easier and less costly than others (cadets/ scouts/ D of E for basic camping and hillwalking stuff, university clubs and local clubs) with some being more accessible to people with more privileged upbringings. I enjoyed the benefit of some of those which probably weren't open to you (cadets/ D of E at school, having time to be a scout leader for a bit of my adult life and a university club).
The basic kit is accessible if you want to try it out (i.e. hiring from a wall) but there are spikes in cost when you decide to keep climbing (buying boots, chalk bag, harness and belay plate), progress outdoors and start leading independently. Those focused on walking have similar spikes in cost too (progressing into the mountains, starting winter stuff and starting in multi day stuff). There is also the cost of learning how to do those things from someone else unless you are in a club or have a friend willing to teach you and being able to access places you can do your bits of climbing/ mountaineering.
Discrimination on class based grounds includes high costs (kit, training, travel and accommodation if travelling a long way) and social barriers (attitudes including the ones mentioned by Rick Graham, not knowing people who can advise you on getting started, difficulties committing to regular participation if working shifts). The issue of cost is relevant to most diversity related stuff given the existence of gender/ race/ sexuality pay gaps, underemployment of disabled people, and so on, social barriers apply to all of them too, but with different aspects applying to each.
Mountaineering clubs can make a massive contribution to the accessibility of the hobby (and in the past did so very successfully, as can indoor walls by being close to where people live). There is some stuff around reducing training cost with the Ready to Rock courses and the Conville courses etc, however this could go further, more formal training of novices by club members could also contribute as it does with clubs in many other sports, but that needs funding to train people willing to act as voluntary instructors.
I doubt that the routes into climbing used by Brown and Whillans would look the same now as they did back in the 50s.