An air carrier First Officer failed to select the MCP Altitude Capture during a descent to an ILS and subsequently descended below the glide slope which caused ATC to issue a low altitude alert.

Narrative

Descended too soon inbound to JFK via LENDY 5 arrival; then vectors for VOR/DME 13L. Approach Controller was giving final speed adjustments to us; when midstream he said Tower just advised that a layer was moving in and we were now cleared to 'stay where you are for now' (descending to 2;000 FT) and 'you're cleared to cross Canarsie at 1;000 FT (which are the lower altitudes allowed on the approach when cleared by ATC) cleared approach; contact Tower 119.1 good day.' So the Captain; thinking we were closer in then we actually were; reached up to change the altitude in the MCP window; while I was reaching cross cockpit to adjust speed and continue slowing. So the airplane no longer had a 'floor' of 2;000 FT at which to get Alt Cap and Alt Hold instead we kept descending in FLCH so as soon as I detected we were not level nor slowing as swiftly as I was expecting; I'm cross checking speed/distance and note we should be higher 'out here' and immediately arrested the descent with V/S vertical speed mode. As we're doing this the Tower calls us and says altitude alert so I then initiate a climb to get back up to profile. We were over the Van Wyck Expressway on the Canarsie arrival and could see through the numerous holes/patches in the clouds so we verified our position and continued the approach without any further complications. To prevent recurrence; get down sooner; drag it in (which is hard in New York). That way last minute changes would occur while the aircraft is already in Alt Hold or better adherence from crewmembers to SOP which would have included stating the change to the MCP altitude window.