Stephen was born, with absolute certainty, after 1 Jan 1626/7 and before 1 Jan 1648/9.  His wife, Dorcas, was baptised in late 1633 and was probably born earlier in the year.  The average married man is four years older than his wife, so a birthyear ca. 1629 is reasonable for Stephen.   This date has ramifications because Stephen's father is believed to have immigrated in 1634 — and certainly he was in Massachusetts by 1634.  If Stephen was born in 1629, he was born in England and his father must have been married in England.  One often sees either 1634 or 1637 given as the date of Stephen's  birth, but these appear to be estimates based on the writer believing that John and Jane met and married after he came to America.  Of course, it's possible for Stephen to be younger than his wife and to have been born in America, but not probable.

1.  Timothy Hopkins.  1932.  John Hopkins of Cambridge, Massachusetts, 1634, and Some of His Descendants. Stanford Univ. Press, Stanford, CA.

2.  Gary Boyd Roberts.  1995.  Ancestors of American Presidents.  New England Historic Genealogical Society, Boston.

3.  Sylvester Judd.  1851.  "Ancestors of Rev. Samuel Hopkins, D.D., of Newport, and Their Children." New England Historical and Genealogical Record.  5(1): 43-45.  On p. 43:

Stephen Hopkins, only son of John, married Dorcas Bronson, daughter of John Bronson, of Farmington, and resided at Hartford.  He died in 1689, and his widow in 1697.  He names, in his will, six (sic) children, viz.:  John, Stephen, Ebenezer, Joseph, Dorcas, Webster, and Mary Hopkins.