The snag was that he already had three girls, none of whom was inclined towards stardom, and his wife, Oracene, was not keen to have more children.
So he hid her birth control pills.
Not one to do things by halves, Williams decided to double his chances after Venus was born. This time he persuaded one of his wife's friends to take the birth control pills from Oracene's handbag. Along came Serena.
At the United States Open on Saturday, Serena Williams, aged 17, became only the second African-American female to win a Grand Slam singles championship. She received a cheque for $750,000 (pounds 475,000) and a telephone call from President Clinton after defeating Martina Hingis, the world No 1, who had beaten Venus, aged 19, in the semi-finals.
From the time Venus and Serena first learned to play on park courts in Compton, Los Angeles (their father having made peace with members of the Bloods gang, who were not averse to shooting at tennis players), Richard Williams predicted that the younger daughter would be the better player.
He is not always right. Two weeks ago, he forecast that Venus and Serena would meet in the final. Still, one out of two is not bad. And Venus did contribute to her sister's success by draining Hingis of much of her mental and physical energy in their semi-final on Friday evening.
When the gigantic Arthur Ashe Stadium was opening, in 1997, Hingis defeated Venus in the final. Serena not only settled a score for the family in an arena named after the only African-American man to win Grand Slam singles titles, but she also followed the pioneering example of Althea Gibson, who won five Grand Slam singles titles in the 1950s, the last of them at nearby Forest Hills in 1958.
No sooner did Saturday's cheering for Serena's 6-3, 7-6 win stop, than regular observers of the game began to wonder if the triumph would have an adverse affect on Venus. The two sisters could not be closer, and nobody cheered louder than Venus. They practise together, play together, pray together, and share a home in Florida.
Until now, Venus was the front runner. She defeated Serena in their only previous singles final, at the Lipton Championships in Florida in March, and remains one place above her at No 3 in the world rankings. But little sister has won the big one.
"I think, if anything, it's going to motivate Venus," Serena said. "She was up there supporting me. After I lost my first two match points, I looked over - to not my mom or dad, I saw them also - but I saw Venus over there, really making sure, pumping me up. It really helped me.
"Venus was really down after her match with Martina. I've never seen her that down before. That encouraged to be even tougher out there today.
"My dad used to say to Venus and I: `Which big ones do you want to win?' I said the US Open. Venus said Wimbledon. I guess when she wins Wimbledon, she going to have the same feeling that I have now. It's really exciting."
Venus Williams was not the only one masking conflicting emotions with smiles, and joining in the hugs of congratulations.
Hingis achieved as much for herself, if not more, with her gracious manner in defeat as in any of her prodigious victories. The quality of the 18- year-old Swiss's play in advancing to the final had already erased doubts that she would be able to raise herself after the disappointment of a first-round loss at Wimbledon, shortly after the temper tantrums associated with her defeat by Steffi Graf in the French Open final.
Although struggling from the start to stem the power of a second Williams sister on consecutive days, Hingis fought to make her court-craft count.
After losing the opening three games of the first set, Hingis had two game points to level to 4-4. In the second set, she recovered from 2-3 and 3-4, saved two match points at 3-5, and broke Williams to love as the American betrayed nerves for the first time in serving for the set at 5-4.
The match appeared to be moving towards a third set when Williams double- faulted to 0-30 with Hingis leading, 6-5, but Williams contrived a backhand drop-shot off a reasonable return for 15-30. Hingis had one set point, after the second of five deuces, only to net a backhand under pressure.
Hingis made another recovery in the tie-break, from 2-4 to 4-4, only to be cracked by a mighty forehand return of her second serve. Hingis then played a backhand over the baseline on Williams' third match point.
Told that Richard Williams said she had been "scared out there", Hingis said: "I think I wasn't the only person at the end who was scared. Serena had two match points and wasn't able to close them out. I think she was a bit more scared than I was, actually, at the end, because I've been there, done it.
"She's a great competitor, a great fighter. It paid off for her today. Hopefully, next time I'm going to have a better chance."
Two of the most remarkable statistics from the final was that Hingis produced only seven winners, including serves, and that Williams committed 57 unforced errors. "Wow," Serena said. "Imagine if I stop making those errors."
Althea Gibson, the tall, athletic forerunner of the Williams sisters, lives in semi-reclusion in East Orange, New Jersey. "One of her best friends told me she wanted to see another African-American win a Slam before her time is up," Serena said. "I'm so excited I had a chance to accomplish that while she's still alive."
Angela Buxton, a Wimbledon singles finalist in 1956, keeps in touch with Gibson. "She was looking forward to watching the final on television, and is pleased that it keeps her name alive. Pleased in a way, because deep down, Althea still believes she was the best."
She was certainly the first, in that pre-open era, and she did not win a dollar.