Looking for birth records pre1875

Hello, I just figured out that my McNamara (also known as Mack) family had lived in Hamilton County, OH.
Father: James Mack (as in 1860 census) born Ireland about 1829
Mother: Mary also born in Ireland about 1835
Children were all born in Ohio:
Margaret ~1855
James ~1860
John 2 Feb 1864
Anna ~1867

They lived in the 16th Ward of Cincinnati on the 1860 and 1870 censuses.They were in Indiana by 1880.

How can I go about finding any information on their births?Mostly John since he is a direct ancestor.

Thank you!
Pam