NYPL subscribes to a number of databases that provide cardholders with access to digitized, searchable historical newspapers. These databases include both major, mainstream national newspapers and smaller publications serving a specific ethnic or political group. Both national and local news outlets frequently covered labor stories, which can be found in the below databases using keyword and advanced search. Try searching keywords like the name of a union or local, "labor" (or a set phrase like "Chicago labor"), and restricting your search to specific years or dates.
In addition to the below, free online resources like Chronicling America, Google Books and HathiTrust Digital Library are immensely helpful for finding full runs of union periodicals that are in the public domain, including ones digitized by the New York Public Library. HathiTrust advanced search allows users to search by LCSH. Try using the tips in the "Trade Union Materials by Industry" tab to find materials.
Other libraries have also created useful online resources for accessing historical labor-related periodicals. See the University of Washington's Labor Press Project, New York University's Digital Tamiment and Cornell University's Kheel Center Digital Collections.
Historical and other research materials from the NYPL Business Center, formerly known as the Science, Industry and Business Library (SIBL), have been relocated to the Stephen A. Schwarzman Building. Among these are a number of trade journals representing an extremely broad range of industries. These publications can provide useful historical context about the industry you're researching and may also contain news about unions and labor relations.
SIBL staff wrote a six-part series of blog posts highlighting historical trade journals that have been digitized.
The Library also provides access to current trade publications. This guide, also compiled by Business Center librarians, contains a list of currently-received trade publications in print, organized by industry. See here for tips on finding online trade journals.