Only five operators in Britain scored more than two out of eight points in analysis by the Campaign for Family-Friendly Trains.
Thirteen firms were assessed on factors such as the availability of dedicated space for unfolded pushchairs, step-free access, baby changing facilities and information for families.
The lowest totals were awarded to Avanti West Coast, CrossCountry, Grand Central and ScotRail, with one point each.
TfL Rail – which will be rebranded the Elizabeth line from May 24 – earned the highest score with 3.5 points.
Ten operators such as Great Western Railway, Northern and TransPennine Express failed to respond to requests for information, which the Campaign for Family-Friendly Trains suggested shows they have “little concern for the issues”.
The group wants trains to have areas for unfolded pushchairs to help passengers travelling with young children.
It claims uncertainty over the availability of these spaces is “frequently cited by parents as a barrier to train travel”.
No operator was found to have a dedicated area for unfolded pushchairs.
Campaigners also want rail firms to make it easier for passengers to get on and off trains, and obtain information relevant to families.
The report concluded: “Overall, train operating companies across the UK are failing dismally to support families with young children.”
Alice Delemare Tangpuori, a spokeswoman for the group, said: “We hope this annual scorecard encourages train operating companies to continue to go the distance for families travelling on their services.
“As leisure and discretionary travel becomes more important for the railways, creating family-friendly trains must be a priority route to attract more passengers.”
Jacqueline Starr, chief executive of the Rail Delivery Group, which represents operators, insisted the industry wants “everyone to be able to benefit from travelling by train”.
She added: “We value Campaign for Family-Friendly Trains’ feedback and are in frequent conversation with them and with Government officials.
“We want to make the railway more accessible to all and we’re reviewing what changes we could make in the short term to improve journeys for families.”