In St. Joseph New Cemetery there is buried John Walsh, interred 7/13/1893, 17yrs, parents James and Mary. Buried in Sec. 3, range 13, grave 14.
There is also an infant Walsh, stillborn, interred 4/4/1874 in St. Joseph New, parents James and Mary, sec. UCG.
St. Joseph New is very accommodating re sending copies of their burial information; a call is likely all that is needed..

Irish Catholics during the period attended either St. Edward Church, or St. Peter in Chains Cathedral. I have read that the pastor at St. Edward's did not keep very good records. St. Peter in Chains Cathedral is still extant; they may have records, unless they are all in the Archdiocesan archives, which is closed at present.

Procter & Gamble's Ivorydale factory was/is in St. Bernard, Ohio; the Walsh family may have moved to St. Bernard, or at least closer to the factory. St. Bernard is a separate city from Cincinnati and likely has its own records. There are also two Catholic cemeteries in St Bernard - St. John Cemetery and St. Mary Cemetery; they mainly catered to German people, but perhaps some Irish were buried there. It would be worth checking.