Barnes is lovely, great schools, international but largely white. Also expensive and transport isn't the greatest. Beware that hammersmith bridge is expected to be rebuilt so closed for 6 months and it is a flood zone.
Putney is more mixed financially and racially. It has cheaper areas away from the centre. But then primary schools are a problem, even religious ones. Better transport links. Lower council tax.
I suggest you think about the primary school options and demographic of those rather than the area.
For example. Putney's Catholic and c of e schools are massively oversubscribed and clustered with a state primary so The catchment areas are all centred there. The other outstanding state primary had a catchment of 250m or so! Further out (typical first house areas) have somewhat mixed intakes of people that didn't get into those outstanding schools, or parents go private. So social segregation based on income.
Barnes has great primaries, but where you live defines the school. If you want to get the tube from Hammersmith then Lowther school covers the North area. Probably the most mixed school economically and racially, possibly the least desirable in the area, but still a good school.
Long term secondary schools are a bit of a gap, due to the private school split. If you can afford private this may not be an issue. For the others Richmond park academy is improving, but who knows what the catchment will be in years to come?