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Manhattan hotels can be expensive and so can travel in and out of Manhattan, so here is a list of the best areas (outside of Manhattan)for inexpensive hotels close to public transportation, and then the bottom few links are for the transportation modes to and from Manhattan.
The NY City Subway sytem is the least expensive way to get into Manhattan from the outer bourroughs. the last stop on each train line in Brooklyn and Queens is about 1 hour away from Manhattan, so the fewer subway stops away from Manhattan you are the less time you will spend traveling. The Closest nieghborhoods in Queens and Brooklyn (as said) are LIC in Queens and Williamsburg in Brooklyn. Thes two nieghborhoods are the easiest to get to, and travel can take as little as ten minutes.
If you are staying close to the river this is a fun transport to Manhattan, instead if the subway (or NJ PATH from Jersey City). For LIC section of Queens and Brooklyn you would chose the "EAST RIVER FERRY" and for Jersey City you would chose the "HUDSON RIVER FERRY"