The red circle with a slash is now accepted as a general prohibitory
symbol. What is the origin of this symbol? I recall James Burke
mentioning in one of the "Connections" TV shows that it was based on
the mark of a powerful european family but have not been able to
confirm this or learn more.

I have not been able to find any mention of a specific European family
in connection with the circle/slash symbol. Perhaps another Researcher
will be able to determine the exact source of James Burke's reference.

Here is an excerpt from an interesting article which traces the
universal prohibition symbol to a possible origin in heraldry:

"Ø", the universal symbol for "don't do this" is taken from the
medieval art of heraldry... In medieval times the device emblazoned on
the shield of a knight was more than a badge of identification.
Heraldic coats of arms were pictographic advertisements for the
knights who bore them on their shields. Generally, a knight who
inherited (or bought) his title would have a horizontal bar across the
top of their coat of arms while a knight employed by the church
displayed a cross dividing it into quarters. Knights of the sword,
however, who had earned their title in battle, had a diagonal slash
across their coats of arms in imitation of the baldric upon which
their sword was hung. The right-handed knight of the sword, therefor,
proudly displayed a diagonal bar on his shield that ran from upper
left to lower right, in imitation of his baldric.

Left-handed knights reversed this device to indicate their preferred
fighting stance... The fact that a left-handed foe was more difficult
for a right-handed fighter to defend themselves against, together with
the superstitious belief that left-handedness was a sign of demonic
possession, led to the nomenclature for a reversed coat of arms. It
became known as the "bar sinister." To a knight at a tournament, the
bar sinister on another knight's shield carried a message of ominous
treat, of danger and evil.

...In more recent times, a circle with a slash from upper right to
lower left has been used alone to indicate danger or placed over an
icon to indicate something forbidden. The most common example would be
the letter "P" under a null sign to indicate "no parking." This symbol
is common in all European countries where knights once fought.

In amerika, however, where the history of heraldic symbolism is
studied little, if at all, the sign for "don't do this" has been
mistakenly interpreted as a circle with a diagonal slash in either
direction.

BobGod.com
http://www.bobgod.com/oldendaze/poli18.html

It should be noted that the official, ISO-recognized prohibition
symbol is a red circle with a diagonal slash that runs from upper left
to lower right. In heraldic terms, this is known as a "bar dexter."
The presence of a bar dexter or a bar sinister on a coat of arms may
indicate legitimate or illegitimate birth in a family's lineage. For
more on this, see the descriptions and the images on this page:

Bar Sinister
http://www.bar-sinister.com/about.htm

You might be interested in my answer to a previous question about
prohibition symbols:

http://answers.google.com/answers/main?cmd=threadview&id=66657

Bar Sinister - - Also carrys another meaning.  It is the sign of the
'Bastard.'  In our family history, (without giving away my identity
here) there was a time two centuries ago when the eldest son had to
marry first.  As it happened, a younger son married before the eldest.
 He kept the family name but had to leave the country and settle
elsewhere.  He was also allowed to keep the "arms" but with the change
from Bar Dexter to Bar Sinister.

So the term 'bastard' and Bar Sinister when applied to 'arms' does not
always mean "illegitimate birth" in a family history.  It can also
mean a marriage which takes place outside the proper and accepted
order.