Added: Shelly types quicker than I do! 
Hi Richard: Hope this answers your question. I use the SSDI on Rootsweb, which is at
http://ssdi.rootsweb.ancestry.com/cgi-bin/ssdi.cgiThere is also a version on FamilySearch.org (at the bottom of the main menu), but it does not seem to have middle initials.
On Rootsweb, you can search by place of issue by clicking Advanced Search on the search page, then enter a name and use the scroll down Issue menu, just above the Submit button, to limit the search to a particular state.
The place of issue of a person's Social Security number is given in the last column of the entry on the Social Security Death Index. So in Helen B. Kruk's case, it would be Connecticut
HELEN B KRUK 10 Apr 1912 18 Aug 1991 (not specified) (none specified) 047-03-3456 Connecticut
One thing to bear in mind about the SSDI: It mainly covers deaths that occurred after 1962. If someone died before 1962, he or she is likely (but not definitely) not to be in the SSDI.
From the Rootsweb SSDI. Three entries for Alexander Kruk:
The V in the first entry means that the death was verified by a family member. Bold face items explain the entries.
ALEXANDER KRUK 24 Oct 1915 26 Aug 2000 (V)
Place of death: 11003 (Elmont, Nassau, NY)
Place of last benefit: (none specified) 088-05-4258
Issued in New York
ALEXANDER KRUK 17 Dec 1889 Jul 1965
Place of death: (New York
State?)
Place of last benefit: (none specified) 108-07-7768
Issued in New York
ALEXANDER H KRUK 21 Nov 1918 25 Jan 1996
Place of death: 18508 (Scranton, Lackawanna, PA)
Place of last benefit: (none specified) 170-09-0697
Issued in Pennsylvania
There were five entries for William Kruk.
WILLIAM P KRUK 17 Dec 1919 11 Feb 2005 (V)
Place of death: 11040 (New Hyde Park, Nassau, NY) b]Place of last benefit: (none specified) 065-16-1139 New York
Elmont and New Hyde Park, NY are suburbs of New York City on Long Island. They are a few miles apart (which is why I added William Kruk. Nassau County borders the New York borough of Queens.
I would be very careful in assuming that any of these are your folks. A search for Kruk in the SSDI returns 155 entries where the Social Security number was issued in New York State. There were 58 Kruks in a New Jersey search.
Regards,
John
