BACK IN the thirties, when people sang their way through the Great Depression, and an aspiring actress could get off the bus from Utah in the morning and star in a Broadway show by night, you knew your true love the moment you met him--or so Dames at Sea would have us believe. Nothing's wrong with that; any show where the piano on stage keeps playing after the piano player's stood up to sing knows just how silly it's being. Dames at Sea, written in the sixties, spoofs thirties musical comedy conventions in a finely polished and very funny production. But one sequence leaves a bad taste: outdated assumptions are appealing as long as the parody is obvious.

Everyone except for six-year olds who've seen "Annie" knows that the thirties weren't all that happy. George Haimsohn and Robin Miller, who wrote the book and lyrics of Dames at Sea, surely do--but did they know what they were doing when they inserted a particularly offensive quasi-Oriental song and dance into their show? "Singapore Sue" (She's so soft and gentle/She's my favorite Oriental) plays on thirties fantasies about sinister Orientals: the pantomime that accompanies the song portrays a wicked slant-eyed madam shanghaiing girls to work in her brothel, where they lure Occidental sailors to their doom. The Depression was racist as well as naive, in the view Dames at Sea presents, and the former attribute is not particularly laughable. After the Asian-American Association protested a similarly offensive stereotype in last spring's Pudding Show, it is astonishing that the Hasty Pudding Theatricals, which oversaw the production of Dames at Sea, did not cut this sequence, which is as irrelevant to the plot as it is unpleasant.

Except for those five minutes of the show, Dames at Sea is an almost unobjectionable production. From the moment little Ruby from Utah walks into "any 42nd Street Theatre" and announces she wants to star in a Broadway show, Ellen Zachos' wide-eyed insouciance insures Ruby will make it big in show business. But first the nasty leading lady of the show about to open that night, Mona Kent (Susannah Rabb), must be eliminated, done neatly when the public-works projects of the W.P.A. force the play-within-the-play to open on a battleship, where Miss Kent succumbs to sea-sickness. And of course Ruby has to fall in love along the way; one of those shore-leave sailors who always stroll along Broadway (Web Stone) appears just as she faints from hunger to catch her and croon in her ear, "It's not Greta Garbo, it's not Jean Harlow, it's you, it's you." She regains consciousness in time to join him in a charming duet. Zachos and Stone both sing exquisitely and act almost as well--stagily and mannered, just the way they should.

There's less acting to be done: this musical is very musical. With eight songs in each of the two acts, the dialogue never lasts long enough to be dull. Except when Howard Cohen's on stage. Fortunately, he has the smallest part of the six leads, although he doubles for the director of the Broadway show and the captain of the battleship. Cohen sings passably, but none of his spoken lines carry any conviction, And an actor needs conviction to get away with lines like "It's a chance in a million, but it just might work!" But the rest of the cast outweighs Cohen's weakness.

Maggi-Meg Reed superbly plays the wisecracking chorus girl who takes Ruby under her wing (a resemblance to Joan Blondell certainly helps). The chorus-girl's argumentative romance with yet another sailor (Frank Pastor) makes Ruby's saccharine affair a little less cloying. Although conducted in a duet which dreamily imitates the worst of Cole Porter, Mona Kent's seduction of the wealthy Captain injects some welcome opportunism into an otherwise hopelessly unworldly world. Goodness prevails, but, gratifyingly, even the selfish end up happy in Dames at Sea.

MONA KENT deserves her happy ending; Susannah Rabb maker even selfish bitchiness appealing. Rabb isn't as good a singer as Zachos or Reed, but David Edelman and Penny Outlaw, the musical directors, have cleverly arranged her songs so she can talk her way through the hardest parts. After all, it's in the script that the ingenue should sing better than the fading star who got where she is on her sex appeal. Rabb brings a glorious cattiness to her role--drawling "I'm sure" when poor Ruby stammers out her name--and dances with the proper verve.

Amateurism is easier to spot in dancing than in singing or acting, but the clever choreography, done by Cyndi Silva, keeps Dames at Sea moving without overburdening the actors. And this earnest, lip-biting, Ruby Keeleresque style suits the script and the spirit of the show.

That spirit is infectious: it's fun believing in happy endings for an hour or two. Enjoying Dames at Sea means accepting the show's premises; raising no eyebrows when a song interrupts a conversation and sighing along with the lead when he looks into his beloved's eyes and assures her, "You're the only one for me in this screwy world."