Il codice è questo a me sembra a posto
Private Function controllanome(ByVal nome1 As String, ByRef dbConn As ADODB.Connection) As Boolean
Dim dbRec As New ADODB.Recordset
Dim strSQL As String
strSQL = "SELECT * FROM Staff WHERE Cognome = '" & nome1 & "'"
DoEvents
Set dbRec = dbConn.Execute(strSQL)
controllanome = Not dbRec.EOF
Set dbRec = Nothing
End Function