Former president Gerald Ford, his wife Betty and their daughter Susan enjoyed a quiet New Year's celebration, ushering in 1988 with a toast of sparkling cider. Accompanying them were members of the nursing staff at the Eisenhower Medical Center in Rancho Mirage, Calif., where Mrs. Ford is recuperating from recent surgery to repair sutures on her chest.

Mrs. Ford followed up the celebration with a New Year's Day stroll around the hospital corridors. A hospital spokesman says she may be released this weekend, although she had to cancel plans to join the Reagans at tonight's opening of the Bob Hope Cultural Center in Palm Desert. Starting With Hope

The social event of the year, so far, has got to be this evening's glitzy debut of the Bob Hope Cultural Center, a $20 million, privately funded, 1,166-seat theater that will feature popular entertainment and provide space for community performing arts groups. Hope, just back from entertaining the troops in the Persian Gulf, will be on hand to dedicate the facilities.

The Hollywood contingent includes: Lucille Ball, Kirk Douglas, George C. Scott, Mary Martin, Nancy Sinatra, Diahann Carroll and Yakov Smirnoff. And from the political world: the Reagans, California Gov. George Deukmejian, White House Chief of Staff Howard Baker, national security adviser Colin L. Powell and CIA Director William Webster.

The evening's entertainment will include music by pianist Van Cliburn, songs from Andrew Lloyd Webber's new musical "Phantom of the Opera" and a filmed tribute to Hope that includes bits from Johnny Carson and former president Ford. It will be videotaped for inclusion in the next Bob Hope special. Princely Benefit

Rock star Prince and jazz trumpeter Miles Davis jammed for charity on New Year's Eve. At an invitation-only, $200-per-person fundraiser held at Prince's new Paisley Park studio in the Minneapolis suburb of Chanhassen, the two played for three hours to benefit the Minnesota Coalition for the Homeless.

At midnight Prince sang the traditional "Auld Lang Syne." Then Davis left his table to join him in a long encore that began with "It's Gonna Be a Beautiful Night."

"You all expect an awful lot for your $200," Prince told the crowd at the end of the performance. Schwarzenegger's Home Role

Movie star Arnold Schwarzenegger concedes he has slowed down some since his marriage to newscaster Maria Shriver, daughter of Eunice Kennedy Shriver and Sargent Shriver. "I do whatever my wife tells me," Schwarzenegger told the Boston Herald recently. The well-muscled actor claims that washing the dishes, making the bed and using a feather duster don't bother him since he was never really a "macho type guy." As he explains it, "It's only in the movies that I'm a gutsy tough guy." Words to Purge

Now that 1987 is over, we may have seen the last of "safe sex." The expression, that is, not the practice. The phrase should be given a life prison sentence with Dr. Ruth Westheimer, the syndicated sex adviser, according to the Unicorn Hunters of Lake Superior State University. The nationwide group of thousands of writers, poets, students and teachers has just published its annual New Year's Dishonor List of Words Banished From the Queen's English.

Other hackneyed phrases these word police would like to see hacked from daily use include "cutting edge," "on a roll" and "alternative life style."