Transportation in Morningside Heights before the IRT
The area of Morningside Heights was not completely without mass transportation before the 1904 opening of the IRT subway, although it was not very well-served. The options included 2 forms of mass transportation which were already widespread in New York: an elevated railway line (or "El"), and a network of surface streetcars, including lines along Broadway and along 125th Street. Brian Cudahy points out that the surface streetcar networks, because they had to compete with other street traffic, could "…scarcely be called true rapid transit." There is a map showing the streetcar lines of the Third Avenue Railway Co, including the lines along Broadway and along 125th St, at http://www.nycsubway.org/wiki/TARS_Route_Map.
The "Els" were trains on elevated tracks, which, in the period from the 1870s through 1900, were operated by miniature steam-powered locomotives. They were noisy, sooty, and the cinders from their engines sometimes set building awnings on fire. After 1900, they were electrified. The closest line to Morningside Heights was the Ninth Avenue line, which ran uptown along Ninth Avenue and then, at 110th Street, made a dramatic curve over and on to Eighth Avenue. There were stops at 110th, 116th, and 125th Streets. Originally consolidated and operated by the Manhattan Railway Co., in 1903 they were acquired by August Belmont and merged with the Interborough Rapid Transit Co. (IRT), which was at that time developing the first extensive subway system in New York. Fig. 1 shows a detail for northern Manhattan, from a route map of the IRT from 1904, showing the new subway line (in red) and the existing former Manhattan Railway Co. elevated railway lines (in green). The dotted red lines represent "viaduct portions" of the route, which were above-ground. A high-resolution scan of the full map is available at this link: http://nycsubway.org.s3.amazonaws.com/images/i17000/img_17506.jpg (from the very useful collection of digitized historical subway maps at nycsubway.org). A very clear older route map, from ca. 1881, showing El service in Manhattan before the IRT, is available here: http://www.loc.gov/item/98688705 on the Library of Congress web site. Note that according to the schedule information presented there that some trains didn't run on Sundays, and that service was limited during some hours of the day on certain lines.
Fig. 2 shows a view of the Ninth Avenue El, from ca. 1902, in a dramatic panorama in front of the construction of the Cathedral of St. John the Divine  (for the exhibition about St. John's on this site, click here.)
The view here (looking roughly to compass-north; north-west by NYC grid orientation) shows the tracks running along 110th Street, after they have made the curve from Columbus Avenue (Ninth Avenue further downtown), headed over to resume an uptown path on Eighth Avenue. Note the steam puffs from the locomotive headed downtown. Also, note that although this line was somewhat near the Morningside Heights neighborhood, it certainly didn't run through the center of it - several blocks' walk was required, and there is a considerable natural barrier, in the form of the sloped terrain of Morningside Park, visible in front of St. John's. Access to the Ninth Ave. El's 116th St. station from Morningside Heights was via a set of steep stone stairs, descending through the center of Morningside Park. So, all in all, not a very convenient point of service for Morningside Heights. Lastly, note the large tree-covered areas along the western reaches of 110th St (to the left), indicating that much development had yet to occur.
 Brian Cudahy, Under the Sidewalks of New York (New York: Fordham University Press, 1995), 13.
 "New York City: Morningside Park & Cathedral Heights - showing elevated railroad. Cathedral constr. in background" (ca. 1902), http://www.loc.gov/pictures/item/2003678100/ (accessed on July 21, 2015).