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A bit out of date, but this interactive map of step free London is very good
All the London railway stations :
Barnes Bridge is on the South Western national rail network, and it is not an accessible public transport station for wheelchair accessibility due to the steps to platforms. See pictures and further accessible transport detail from links and detail below.
Underground lines : National Rail South West
AccessAble Access Guide :
Accessibility friendly : No.
General description : All platforms have access via a footbridge consisting of 24 steps. Station is inaccessible to wheelchair usersthey are advised to use Richmond. See wiki
More comment : National Rail Map & Guide