The survey was conducted amid efforts agreed to by Russian officials in 2000 to create a juvenile justice system, RIA Novosti reported Friday.
The system is opposed by conservative groups such as the Russian Orthodox Church, which claims it would damage family values.
While more than half, 57 percent, of those surveyed said they supported the idea of juvenile courts in Russia, 71 percent said children's rights shouldn't take priority over those of their parents.
Some 60 percent said children shouldn't be able to take their parents to court.
Only 16 percent of respondents said defending the rights of minors should be a state function, while 76 percent said that was the role of parents and family members.
Russian Orthodox Church leaders last week opposed the possibility that low-income families could lose custody of their children should they no longer be able to support a large family.
The poll was conducted by the state-run VTsIOM organization, which interviewed 1,600 people in 46 regions. The margin of error was 3.4 percent.