As mentioned earlier, we need more information in order to help (and to avoid duplication of efforts).

1. Where have you searched?
2. Have you ordered any death records?

3. Were either born in New Orleans? Early Orleans parish births (over 100 yrs old), marriage and death records are available from the State Archives. You can search the index for each of these, as well as Louisiana Deaths (over 50 yrs old), on the Secretary of State's website.

4. From the Census Records:
I found only one Locksley Borden in 1870 New Orleans (census). The family was headed by Orlando and Polly Borden. This Locksley was born about 1859.
Do these names sound familiar?

Another Locksley (born abt 1886) can be found in Orleans in the 1900's, beginning with his WWI draft registration record. This appears to be the one you are researching - since the nearest relative was listed as "Josepha" Borden.

5. Obituaries very important records - some Orleans parish obits are available on the Louisiana genweb archives site (must do a search).
An index to some newspaper obits are on the New Orleans Public Library site - Josepha Borden is listed, along with her death date and age (1927).
You can order a copy from NOPL.

Search for Locksley and each of his sibling's obit and follow the trail for mention of parents' names.

For example: One obit mentioned parents Jessie Sr. and Olivia.
Possible?
If so, then marriage records could reveal Olivia's surname.

Without more info, it will be difficult to provide much more assistance.

Good news: Orleans parish is one of the easier Louisiana parishes for research.