The original manuscript clearly reads the day. When Oliver Cowdery initially copied this passage into the printer’s manuscript, he wrote “at the dawn of day”; but then he virtually immediately corrected his error by inserting the definite article supralinearly (there is no change in the level of ink flow). Of course, in modern English we expect “at the dawn of day”, without the definite article the in front of day. This expected form of the phrase also influenced the typesetter for the 1879 LDS edition, who omitted the the here. The LDS text has continued with this shortened reading. For another example of the tendency to omit the the in “of the day”, see under Jacob 6:5.
Elsewhere the Book of Mormon systematically has the definite article in phrases like “of the day”, “of the morning”, and “of the night”:
As David Calabro points out (personal communication), the same use of the definite article is found with the phrase “the dawning of the day” in the English of the King James Bible (not “the dawning of day”):
Summary: Restore the original definite article in front of day in Alma 47:14: “at the dawn of the day”.