Hello, I found this sole entry for Alexander Bassett within the 1861 Glamorgan Census for you:
Glamorgan Family History Society 1861 Census CD
4038 1861 CENSUS OF LLANDAFF. REF.R.G.9/4038, E.D.21.
Piece: 4038 Folio: 019 Schedule: 153
Address: NO.1 LLANDAFF PLACE
BASSETT ALEXANDER HD M 36 CIVIL ENGINEER
WIL GREAT CHEVERELL (c1825)
BASSETT EMMA MARY WI M 32 -
WIL COLSTON [?] (c1829)
BASSETT ALEX BARKER SO U 4 -
GLA CARDIFF (c1857)
BASSETT KAT E.M.? DA U 2 -
GLA CARDIFF (c1859)
BASSETT FLORENCE E. DA U 4M -
GLA LLANDAFF (c1860)
BEYNON ELIZABETH SV U 46 COOK
GLA PORT TYNON [PORT EYNON] (c1815)
PUGH ELIZA ANN SV W 32 NURSE
GLA CARDIFF (c1829)
WESTCOTE HARRIET SV U 22 HOUSEMAID
SOM WIVELSCOMBE (c1839)
FREDERICK MARY ANN SV U 16 UNDER HOUSEMAID
GLA LLANDAFF (c1845)
I do not have the 1871 Census
Here is the 1881 entry with Alex B Bassett :
Dwelling: 2 Landscape Pass
Census Place: Llandaff, Glamorgan, Wales
Source: FHL Film 1342272 PRO Ref RG11 Piece 5288 Folio 46 Page 7
Marr Age Sex Birthplace
Helen J. EVANS W 54 F Bristol
Rel: Head
Occ: Annuitant
Elizth. HOWE U 15 F Canton, Glamorgan, Wales
Rel: Serv
Occ: General Servant
Chas J. WILLIAMS U 28 M Winchester
Rel: Boarder
Occ: Organist Of Music Plac Oxford Llandaff Cathedral
Alex B. BASSETT U 24 M Cardiff, Glamorgan, Wales
Rel: Lodger
Occ: Civil & Mining Engineer
In addition here all all the Cardiff born Bassett's in the 1881 census:
BASSETT, Alex B. Lodg <1857> Birth: Glam Cardiff
Census: Glam Llandaff
BASSETT, George Serv <1858> Birth: Glam Cardiff
Census: Glam Cardiff St Mary
BASSETT, Emma Serv <1867> Birth: Glam Cardiff
Census: Glam Cardiff St John
BASSET, Lucy Dau <1873> Birth: Glam Cardiff
Census: Glam Cardiff St John
BASSETT, Rosinna Dau <1880> Birth: Glam Cardiff
Census: Glam Cardiff St Mary
If you see any of interest I can look at the full entries for you.