If we are talking about the national government of the United States, the main way in which it supported unions during the Great Depression was through the passage of the National Labor Relations Act of 1935. This act is better known as the Wagner Act. The government passed this partly because it was a Democratic government and believed in unions and partly because it felt that unionization would help the economy as a whole.
Before the Wagner Act was passed, the government had tended to side with employers against unions. Strikes and unions were often held to be illegal conspiracies. Courts would issue injunctions to prevent strikes.
The Democratic Party was more pro-labor. As a result, when it came to power in the 1932 elections, it moved to help labor unions. It did so in part because it felt that unionization would lead to jobs with higher wages and higher wages would help to combat the lack of demand that was making the Depression so bad. The Wagner Act supported unions. It explicitly gave workers the right to join unions and it gave unions the right to collectively bargain on behalf of employees. It also protected the right to strike and it created government agencies to enforce the rights of labor.
In these ways, the federal government supported unions during the Depression.