I have identified Mrs. H. Bassett as Henrietta Bassett. She appears in the 1870 census of Rochester with two sons, George H. Bassett, age 13, and Edwin Bassett, age 2.
She appears in the 1880 census of Rochester with two sons George Hartman, age 22, and Edwin Bassett, age 12.
Based on this information she may have first been married to a Mr. Hartman and had George and later married Mr. Bassett who must have died before the 1870 census.
No other information is known at this time.
Advertising Card found on ebay.
Rochester Democrat and Chronicle, Tuesday, February 10, 1885
Removal
Mrs. H. Bassett has removed her stamping and embroidery establishment to 100 State street (up-stairs), over Mackie’s music store, where she will be pleased to see all of her old customers, and as many new ones as will favor her with a call.
The Rochester City Directory, 1877