Nonprofit human services providers say they can help solve Connecticut’s budget problems by taking over more state-operated programs, an idea that appears to be gaining steam among some legislative Democrats as well as Republicans. The Connecticut Community Nonprofit Alliance on Wednesday unveiled a proposal to shift developmental disability residential services and mental health and substance abuse treatment programs from the state to the private sector. The group says its plan would save $1.3 billion over three years. The proposal comes as Connecticut faces a projected $1.5 billion deficit in the fiscal year beginning July 1. Connecticut has a system where both state employees and private nonprofit providers deliver state services. While many Republicans have pushed to privatize those programs, more Democrats, including Gov. Dannel P. Malloy, support the idea.