An exciting new (to me) way to SSH
grep101.com
grep101.com
Running a browser app. does not, in any way, reduce the potential for keyloggers or other nasties. And a system with a keylogger or other nastie will be able to watch/collect what you do whether you run a browser app., or run putty.exe directly.
If the system is compromised with a keyloger, it does not help at all if you are using a browser extension instead of a traditional client.