Filezilla is certainly a great multi-platform FTP client.
To bypass Gatekeeper without disabling it, simply download Filezilla from the source (never from download.com or any other untrusted third party), uncompress the application, drag it to "Applications", open your applications folder, then right-click Filezilla and choose "Open". Right-clicking a downloaded application and choosing "Open" bypasses Gateleeper. Therefore, you need to make sure that you are getting the download from a trusted source.
Here is the link to get Filezilla:
https://filezilla-project.org/download.php?type=client
Downloading it from most of the "download" websites will have packaged it with an installer and possibly crapware or malware that you did not intend.
By the way, if you ever need a full-featured FTP client that is a polished Mac application, I highly recommend "Transmit":
https://panic.com/transmit/