According to dictionary.com , "bear down" in a nautical sense means to "approach from the windward direction", which makes perfect sense when applied to a storm. Other uses include "to press or weigh down" and "to strive harder or intensify one's efforts", both of which can also be somewhat applied to a storm: it was approaching, its effects were pressing down on the city (causing outages, delays, accidents, and so forth) and its effects were intensifying as it approached.
(Edit: as MMyers also points out, "to bear down on (something)" generally has the meaning of approaching 'in a threatening manner', which is also very appropriate here.)
In a nutshell, then, the opening sentence from the NYTimes implies that officials could see the storm coming, knew that it was going to be bad, and didn't do enough or act quickly enough in what they did do about it.