Some Ohio County birth recordsinformation

Fayette County Births - Carnegie Public Library - Washington Courthouse, Ohio - full transcription of birth records for 1909-1999
http://www.cplwcho.org/birth_search.php

Greene County - Public Library - Birth Records, 1869-1909
http://www.greenelibrary.info/Collections-and-Resources/Birth-Records-1869-1909.html

The Lawrence Register - this independent volunteer website has Lawrence County, Ohio births from newspaper announcements, and extracted from health department and Probate Court records - births range from 1850 to 1940 (formerly at http://lawco.rayhaas.us/births/index/index.htm)
http://www.lawrencecountyohio.com/node/92

The LDS FamilySearch site has images of Toledo, Ohio Roman Catholic Parish Register pages, including baptisms, for 1796 through 2004. Browse only, not indexed. To search newly added records or if link is broken, see the menu for North America at Family Search Beta.
http://pilot.familysearch.org/recordsearch/start.html#p=waypoint&s=waypointsOnly&c=fs%3A1494476&w=0 (requires Flash and Javascript)

Ohio Births and Christenings 1821-1962 - Family Search has indexed about 861,000 birth and christening records, with listings from nearly every county. Recent records may not be available due to privacy laws. Click here for how to purchase copies of indexed records if you need them before they go online. To search newly added records or if link is broken, see the menu for North America at Family Search Beta.
http://pilot.familysearch.org/recordsearch/start.html#c=fs%3A1680845&p=collectionDetails

Genealogy Trails has user submitted transcriptions of birth and baptism lists and announcements for many Ohio counties, from as early as 1821 up to 1980. Note some birth entries are extracted from obituaries and other death records. You may want to modify this Google site search to ease your search, or browse for new entries starting from the Ohio page at:
http://www.genealogytrails.com/ohio/

Rootsweb's User Contributed Early Birth Records Databases include selected births from Butler, Clermont, Columbiana, Coshocton, Franklin, Jefferson, Lawrence, Lucas, Meigs and Muskingum counties. You can perform a search on surname alone, without specifying the location if you are unsure.
http://userdb.rootsweb.ancestry.com/births

Ohio USGenWeb has some volunteer transcriptions of births online, searchable with the following link which requires javascript:
http://www.usgwarchives.net/search/search.cgi/searchoh.htm

If you wish to use other ways of searching by browsing or doing a search engine "site search", here is the URL for the USGenweb archives for Ohio:
http://www.usgwarchives.net/oh/