Access to the contents of newspapers is in most cases made available through periodical "aggregators" such as ProQuest or LexisNexis. The best way to determine our library's holdings of a particular newspaper is to search EDUCAT by title (limiting to "Only Serials"), just as the best way to determine the Columbia Libraries' holdings is to search CLIO by journal title.
In a very few cases a library may maintain holdings of a newspaper on microfilm, though increasingly digital/electronic access is the norm. In virtually no cases are print newspapers kept by libraries (on the Morningside campus and in general), largely due to preservation and space issues.
A useful listing of online newspaper collections, including ProQuest Historical Newspapers, can be found via the Columbia University Libraries website.
Columbia also provides access to a number of online news resources through the Libraries' E-News page, which features a comprehensive list of print, microfilm; various news and current events databases; and links to sites focusing on news broadcasting in various media.