With all the protest against rock music lyrics finally settling down, Tipper Gore said yesterday that she has asked Stanley Gortikov, president of the Recording Industry Association of America, for his help. Now that she and Susan Baker won't have to spend time battling musicians at congressional hearings, they are turning their attention to the problem of the homeless.

Gore said she got involved with both issues around the same time, but the music lyrics protest got all the attention. "Though this issue doesn't have the same sex appeal, it's even more important . . . We hope we have the same sort of success as we had on raising public awareness of rock lyrics."

Gore said she was impressed with the Live Aid and Farm Aid benefits, and asked Gortikov if he would join her efforts in aiding the homeless by getting some musicians to raise public consciousness of the problem. Would she accept help from the groups whose lyrics she had specifically protested? "I wouldn't ask them," Gore said, "but I wouldn't turn down anyone who asked." Charles and Diana Met by Protesters

Royal Watch: seven days and counting. Prince Charles and Princess Diana were greeted in Melbourne yesterday by about 50 demonstrators protesting against British rule in Northern Ireland.

The royal couple ignored the protesters, whom they encountered when they arrived for a charity film premiere. The shouts were easily drowned out by the crowd of 5,000 and a brass band, which was instructed by a police officer to play louder.

"We've had no trouble with them in the past; we usually meet before and set the ground rules. They can say what they like on the street -- this is a democracy after all," a police spokesman said. The demonstration, organized by a group called Australia Aid for Ireland, dispersed quietly after the couple entered the movie theater. Queen Elizabeth Tours the Caribbean

Royal Unwatched: Queen Elizabeth arrived in Trinidad and Tobago yesterday on her first official visit to the islands in nearly two decades. The queen arrived from Grenada, where she was unspectacularly received. Wire services report that although Thursday was a public holiday in Grenada, few people ventured out into the drizzle to see the queen. End Notes

Sen. Robert Kasten (R-Wis.) announced his engagement yesterday to Eva Jean Nimmons, an advertising account supervisor in New York City. The wedding is planned for Jan. 4. Said Kasten, "I've known Eva for some time, and I've known she was the right person." It will be the first marriage for both . . .

Hair stylist Robin Weir is overbooked with appointments due to the royal hullabaloo, so British hairdresser Phillip Hodgkinson arrived in town yesterday to help out. After the Washington rush is over, both Weir and Hodgkinson will follow the frenzy to Palm Beach, Fla., to coif the British and American socialites there . . .

More than $1 million already has been raised for AIDS research and treatment by "The Best of the Best: A Show of Concern." Celebrities expected to appear at tomorrow's Metropolitan Opera House benefit are Dr. Ruth Westheimer, Lily Tomlin, Mikhail Baryshnikov, Carol Burnett and Christopher Reeve . . .

Apollo XIV astronaut Edgar Mitchell has been ordered to pay child support for a son born a year ago to former Playboy bunny Sheila Ann Ledbetter. The court decision, made public this week, came after blood tests showed Mitchell was, in all probability, the boy's father . . .

Pat Buchanan, who turns 47 today, got as a birthday present from the White House media relations staff something the director of communications didn't already have: a framed window for his windowless office. Unfortunately, he still won't be able to see out . . .

It must take more than losing his hair to make Andy Warhol lose his cool. Liz Smith reported in the New York Daily News yesterday that while signing books at a party earlier this week, a woman snatched Warhol's trademark platinum wig and fled. The artist pulled up the hood of his parka and calmly continued autographing his new book.