When it's discovered Bulk and Skull are trying to open Rita Repulsa's dumpster, Zack notes this must be prevented, but doesn't say why, presumably because the other rangers know what he's thinking. However, one reason is given: Trini then says it would blow their cover. Kimberley agrees (presumably to the reason as well as the necessity).
This seems odd. Although Rita could reveal the rangers' identities, even if that would be a problem (funny how she never tried to make it so!) surely the main reason to prevent her release is the danger she poses. Admittedly she is weaker than their modern enemy Lord Zedd (hence Tommy's new powers in the same episode), but having Rita to deal with too would be a serious problem. Indeed, The Wedding later proves as much. It's also notable that Zordon's reason to send Kimberley, Jason and Zack after the dumpster was that Rita "cannot be allowed to escape", so their identities don't seem to be his main priority.
So why, of all the things to focus on, did the rangers worry about their identities? Day of the Dumpster and Green With Evil Part V did insist on "keeping" one's identity a secret, but this surely only means self-revelation is punished with a loss of power, not that accidental discovery costs it. Indeed, in The Ninja Encounter Part III Billy doesn't lose his powers when his identity is accidentally revealed, and nor do Kimberley and Tommy when they deliberately reveal their own identities. That might be because Adam, Aisha and Rocky promised to keep quiet, which Bulk and Skull presumably wouldn't because of their ambition since The Mutiny Part I to expose the rangers' identities. But my main reason to doubt rangers could lose their powers due to Rita blabbing is that she never tried that as a strategy.