The woman was, understandably, taken aback by this unsolicited approach. But she was more surprised when the BA man offered her two free tickets from London to Paris if she switched to BA. She accepted the offer.
Other Virgin passengers were not so lucky. In December that same year a retired businessman booked his 17-year- old son on to a Virgin flight from Gatwick to Miami and onward to the Cayman Islands.
Shortly before the flight was due to leave he was telephoned at home by Virgin and told that the aircraft was delayed for technical reasons.
Since this meant that his young son, who was flying solo for the first time, would miss his connection and be forced to stop overnight in Miami, Virgin had rebooked him in a confirmed seat on an earlier BA flight to Miami from Heathrow. This is common airline practice - BA Business Class and Virgin Upper Class tickets are interchangeable.
When he arrived at Heathrow's terminal 4, however, he was told that his Virgin ticket could not be accepted since it was not 'validated' - to do that it would have to be returned to the Virgin shop in central London.
After some polite protesting a BA supervisor was called. In a letter to Virgin recounting the incident the businessman said: 'I was then told in front of others: 'We are in competition with Virgin. We do not like them, they do not pay us when we carry their passengers and we do not like Virgin passengers.' '
Further protests were of no avail. The father and his son were told that 'if you want to complain, complain to Virgin and not us'. The son was obliged to fly with Virgin despite the inconvenience.
According to Virgin the two incidents were connected by a common theme - they were both products of a worldwide strategy by BA to exert its supremacy over its smaller rival. There were different strands to the strategy, ranging from one campaign called Mission Atlantic to others codenamed Hunters and Operation Barbara.
Were today's 'mother of all libel battles' in the High Court between Virgin's Richard Branson and BA's Lord King not due to end this afternoon with an agreed statement, damages and an apology by BA, the jury would have been treated to a great deal more detail about Mission Atlantic - and much else besides.
According to Mr Branson, this alleged poaching of Virgin passengers by BA and its refusal to accept Virgin tickets amounted to 'sharp business practice'.
The Virgin Atlantic chairman's allegations did not stop there. In an open letter to BA's non-executive directors, dated 11 December 1991, Mr Branson claimed that an internal task force had been set up within BA, sanctioned at the highest level, with a remit to wage a 'dirty tricks' campaign against Virgin designed to discredit and damage its business.
He claimed that smear stories were being peddled around the press by Brian Basham, a PR consultant to BA, questioning Virgin's financial standing and Mr Branson's other business activities.
But in the climate of suspicion, fear and perhaps paranoia engendered by the fierce rivalry between the two airlines Mr Branson had different worries. His private houseboat and headquarters in Holland Park, west London, had both been broken into and confidential papers apparently removed. Cars belonging to senior Virgin employees had also been broken into frequently.
Virgin also said it had evidence that Mr Branson's private business discussions were being bugged and that he and other Virgin directors had been tailed by private investigators.
There is absolutely no evidence that BA was behind any of these activities or even that it knew about them. That did not stop Mr Branson posing the question, but Virgin now accepts there is no link between BA and any of these events.
BA refused to answer the allegations in Mr Branson's open letter point by point, contenting itself with a blanket denial of his claims - which, it insisted, were 'wholly without foundation'.
Finally, BA's patience snapped and it accused Mr Branson of fabricating his claims in order to garner publicity. Mr Branson sued for libel. Lord King and BA counter-sued over Virgin's original allegations.
To the media the story was a gift - Richard Branson, the anti-establishment entrepreneur who summed up the buccaneering spirit of the 1980s versus Lord King, Mrs Thatcher's favourite businessman and the man who had saved BA from the scrapheap.
But to the two protagonists it was the start of a long, acrimonious and ultimately costly legal battle. Before a word is even said in court today the combined costs of both sides are put at pounds 3m.
The fact that writs were now flying did not dampen the press's appetite for the story or halt the speculation. Where had the hostility between the two airlines sprung from, what had sparked the alleged dirty tricks campaign and who exactly was co-ordinating it?
It was Virgin's and Mr Branson's belief that the campaign was being partly handled by Brian Basham, a well- known City PR man who had carried out consultancy work for BA's public affairs department before. In his open letter Mr Branson asked whether Lord King, Sir Colin Marshall, BA's chief executive, and David Burnside, director of public affairs, had known of Mr Basham's activities. Mr Burnside is an Ulsterman with a formidable reputation and a taste for bloodsports and politics.
Some suggested the trigger for the BA campaign was Mr Branson's publicity coup in September 1990, when Britons stranded in the Gulf by the Iraqi invasion of Kuwait were airlifted to safety, not by the nation's flag-carrying airline but in a fleet of Virgin 747 jumbos.
Others suggest the catalyst was a more mundane and pragmatic one - Virgin's increasing commercial success in cherry-picking BA's most lucrative long-haul routes and then mounting high-quality competition at lower prices. When BA was stripped of its rights to operate some flights and forced to hand them over to Virgin - as was the case at Tokyo's Narita airport - it merely rubbed salt in the wound.
In any event, from late in 1990 Virgin began to piece together its evidence. Its headquarters, a nondescript red-brick building around the corner from Notting Hill in London, soon began to fill with testimony from Virgin passengers of how they had been approached or poached by BA staff.
Some came direct from passengers, some via Virgin staff overseas and some came personally to Mr Branson - who, as usual, made copious longhand notes in his ever-present red and black ledger.
At the same time Virgin began compiling a formal complaint to the European Commission charging BA with abusing its dominant position on the transatlantic routes and engaging in anti- competitive behaviour.
Former BA staff also came forward and volunteered affidavits allegedly describing how they had been ordered to shred files referring to Virgin. According to Virgin their sworn statements detailed the lengths to which they were asked to go to undermine the smaller, rival airline by introducing selective fare cuts and refusing to handle Virgin long-haul passengers needing to connect to BA's European network of destinations.
Meanwhile, according to Virgin, the BA 'dirty tricks' campaign was advancing on other fronts. In October 1991 a 48-page 'dossier' on Virgin, compiled by Mr Basham, began to circulate among the press.
Although portentously marked 'private and confidential', the dossier amounted to little more than a mish- mash of financial speculation about Virgin, biographical details of its chairman and tittle-tattle about some of its directors.
Some actually thought the dossier rather flattering to Virgin, if inaccurate in parts. But Mr Branson felt differently and declared it was an attempt to create 'a most damaging impression'.
For instance, the dossier referred to a confidential report on security at Heaven, the gay London nightclub owned by Mr Branson, carried out by the security consultancy KAS, an organisation founded by the former SAS officer David Stirling. According to the dossier the report was highly disparaging about the club. And Mr Branson's open letter alleged Mr Basham tried to interest one journalist in running a 'discreditable' story along these lines.
The atmosphere of suspicion was fed by a series of bizarre incidents even though there was no apparent link between them and BA. In September 1991 the daughter of an employee of the limousine company used by Virgin Records was being employed in a solicitors' office as part of a work experience course. During her time there she was asked by a private detective agency in adjoining offices to help out by typing some documents.
The documents turned out to be verbatim notes of a private meeting at Claridges between Mr Branson and the head of Paramount Pictures. Virgin accepts there is no connection between BA and this incident.
In his search for ammunition to use against BA, Mr Branson has tapped many sources. One of them was Sir Freddie Laker, the pioneer of cheap transatlantic air travel who was forced out of business by the anti-trust activities of BA and a number of US airlines.
Sir Freddie offered to fly over and help Mr Branson underline the threat posed by BA by recounting his own experience. Mr Branson eventually declined the offer. He did however, take to heart the general advice Sir Freddie offered: at the first hint of any dirty tricks campaign, holler loud and long as quickly and as publicly as possible, for to delay could be to go under.
BA and Virgin are in court today because Mr Branson followed Sir Freddie's advice to the letter.
The mountain of evidence assembled by Virgin is now destined to remain on file rather than being spread out embarrassingly for all to see in the High Court over the next 12 weeks.
That small blessing notwithstanding, today is unlikely to be one of the red letter days in the annals of British Airways.