The question of whether you can get a passport is equivalent to the question of whether you are, or can become, a Dutch citizen.
Did your mother lose her Dutch nationality...
To that end, the first question is whether your mother was a Dutch citizen when you were born. When your mother was married, Dutch nationality was controlled by the Wet op het Nederlanderschap en het ingezetenschap of 1892. This provides for several ways Dutch nationality can be lost:
I believe that her marriage did not cause her to lose her nationality unless she made a declaration; this also depends on whether she became South African by marrying your father. Article 8a:
De Nederlandse vrouw, die gehuwd is met een niet-Nederlander en dezelfde nationaliteit bezit als haar man, verliest het Nederlanderschap door haar wil daartoe te kennen te geven aan de bij artikel 12a bedoelde autoriteit.
A Dutch woman married to a non-Dutchman who possesses the same nationality as her husband loses Dutch nationality by declaring her will to the authority indicated in article 12a.
She may also have lost her nationality under Article 7, paragraph 1:
Nederlanderschap wordt verloren:
- door naturalisatie in een ander land ...
Dutch nationality is lost:
- through naturalization in another country ...
...through birth abroad and a ten-year residence abroad?
She may also have lost her Dutch nationality on her 31st birthday, through Article 7, paragraph 5 (I have deleted references to Indonesia because they were no longer applicable at the relevant time):
- voor zoveel betreft Nederlanders buiten het Koninkrijk geboren, door, behalve in dienst van het Koninkrijk, woonplaats te hebben buiten het Koninkrijk gedurende tien achtereenvolgende jaren, tenzij de afwezige voor het verstrijken van die termijn aan de bij artikel 12a bedoelde autoriteit kennis geeft, dat hij Nederlander wenst te blijven.
- as far as Dutch nationals born outside the Kingdom are concerned, by, except in the service of the Kingdom, having residence outside the Kingdom during ten consecutive years, unless the absent person gives notice, before the expiration of the term, to the authority indicated in article 12a, that he wishes to remain a Dutch national.
The age of majority in Dutch law was 21 from 1901 until 1988, so, assuming your mother never lived in the Netherlands after 4 April 1981, and never made the appropriate declaration, she would have lost her Dutch nationality on 4 April 1991.
If you have any older siblings born before your mother's 31st birthday, you may want to talk to them about this. They may be Dutch, and if they are they will most likely lose their Dutch nationality, through a similar provision that is currently in effect, on their own 28th birthdays.
For your mother, and any children she has who are already over 28, there are simplified procedures for the reacquisition of Dutch nationality. There is an option procedure available in some cases, and a simplified naturalization procedure available in others.
More information can be found at the Dutch government's page Becoming a Dutch National. Some of these procedures are available outside the Netherlands, as explained in the brochure How can you regain your Dutch citizenship?
I suppose there is a small chance that your mother's regaining her Dutch nationality might help you in your case, but I do not think that it is likely.
Finally, you shouldn't take my word for this. I may have overlooked or misinterpreted some provision of the law. There may be additional facts that I have not taken into account. You can apply for a determination by a Dutch consulate; they will give you an official answer to the question.
The embassy in Pretoria has a consular section, and there is also a consulate general in Cape Town.