Translation:Do it quietly.
• 조용하게 하세요 - [하세요, please do; 조용하게 (adverb of 조용하), in a discreet manner/without making much noise]
= Lit. (Please do) keep the noise down
= Keep quiet (Active)
• 조용히 하세요 - [하세요, please do; 조용히 (adv of & also passive form of 조용하), stay/be quiet, calm.]
= Lit. - (Please do) Stay quiet
= Be quiet (Passive- Stative)
In daily usage, the 2 expressions although grammatically different, have become virtually interchangeable.