Yes, RuneScape private servers are generally safe to play.
The commercialisation of the RSPS scene has, as far as we know, put an end to fake servers with malware. We do however recommend you to keep your judgment sharp. Do not play a RSPS if it does not provide proper information, does not have a website or does not have a community (forum, Discord, …).
Just like any popular website on the internet, some major RuneScape private servers are sometimes targeted by hackers. To protect you from this, these servers are required to comply with certain security standards (including the encrypted storage of your personal information). We are in close contact with the managers of these servers to ensure they meet the security standards.
For more information regarding safety of RuneScape private servers, we recommend you to check this blog article by Falchios.