State-subsidized home-based child care providers in Rhode Island have overwhelmingly voted to unionize. The Service Employees International Union said Thursday that the vote among providers in the state’s Child Care Assistance Program was 390 to 19 in favor of joining District 1199 SEIU New England. There were 539 eligible voters. The General Assembly approved legislation this year allowing certain child care workers to unionize. The vote will allow the workers to negotiate with the state over pay and benefits. Voting began Saturday at various locations.