How to turn on and off Taobao’s message prompts?

It can be turned off.

After logging into Taobao, click "Site Mail" at the top of the page to enter the mailbox. There is a line of gray words below: When receiving new site mail, the system will not play the sound. There is a button next to it. If you don't want the sound, just click "Close". If you want the sound, just click "Play Sound".