Familysearch has just added a new collection, United States Passport Applications, 1795-1925, which should make finding these records much easier. The collection, which is a combination of two different National Archives and records Administration (NARA) collections, has over 3 million images of which 1.9 million are indexed.
A great example of the information provided in the applicant is the record of Albert J. Cohen (shown below). From his application we learn that he was born on 31 Oct 1874 in Cairo, Egypt. He is married to Krisanti Cohen who was born in 1881 in Crete and that they have a daughter Mary who was born on the 27th of November 1903, also in Cairo. Further we learn the family arrived in the United States in August 1894 on the S.S. Rotterdam. The application also states that Albert was naturalized a U.S. citizen on 11 Mar 1902 in Charleston, South Carolina.
As with all databases these passport records are available free of charge at www.familysearch.org