I really dislike using online ftp programs due to the security concerns. I don't think that anyone can really guarantee an online FTP program as being safe, even if someone has used it and never had issues, it doesn't mean that there isn't a potential issue. Remember too that sites get traded all the time, and I wouldn't be surprised if people did buy or build those sites simply to be able to gain back access to servers.
Actually I neither like nor recommend to use such web-based tools; because security is very important to me. I would like it, if it would be recommended my hosting provider.
I would definitely not use something like Net2FTP because of security reasons. However, you could install your own web FTP client on a server that only you have access too and secure it with .htaccess authentication etc. This would be a more secure alternative than using an exisiting public service like Net2FTP for your Chromebook.
why not using native FTP client such as FileZilla or WinSCP? if security is your concern, so avoid to use publically available web based FTP client, however you can install a standalone version of Net2FTP on your hosting/server
The FTP protocol itself is insecure since it's processing credentials and login details in a plain text mode so your best option is to try SFTP or FTPS.
I put my chromebook into developer mode and then was able to run regular linux binaries in a chroot. I didn't like the idea of using most of the web-based tools available from the Google store.
Because you can't install programs it doesn't run on Windows OS everything works in the Chrome browser. So you mostly just use it for Google apps, and need to rely on online apps.
Online FTP clients are never a good option. If you are willing to spend money on this, go to the Chrome Web Store: https://chrome.google.com/webstore?utm_source=chrome-app-launcher and go to the sFTP Client: https://chrome.google.com/webstore/detail/sftp-client-ftp-sftp-ssh/jajcoljhdglkjpfefjkgiohbhnkkmipm?utm_source=chrome-app-launcher-search and it is available for $2.50. If you are not willing to spend money on it, search, 'FTP' and there are plenty of good free extensions, though not as good.
There are (or at least should be) several ways to get around this. If your host have decent software installed, it usually have some kind of web-based file-uploader after you log in to your admin-account. If that's not an option, you could always run some sort of SSH-client (I'm assuming this is available, I haven't looked into Chromebooks, as I see them as useless as the old netbooks from a few years back) and access the environment that way. Also, you could transfer files via existing services if the server you're on provides the ability to utilize dropbox sync or similar - another way would be to use GitHub and sync directly on the server (although if you have confidential information in the files themselves, you'll need to buy a paid account at GitHub to be able to mark the project as private) - this will still need a way to connect a command line (SSH) to the client. Or, as @freelancewebaz said, put your chromebook into dev-mode, and run proper linux binaries
Is it secure to use an online web-based FTP client to upload your files to the website host? I’m not positive if such an FTP client exists unless you talking about the file manager services produced by most cPanel hosting services. Any time you have FTP login credentials saved online somewhere there is a risk. Why not keep things local and more in control?