I know that SSH from the command line is easy enough, but would like to give my students that use OS X a GUI option.
Is there a PuTTY equivalent for the Mac?
The best GUI application for SSH (and everything else you can do on the command line) is iTerm 2. While the original iTerm had a tabbed interface before Terminal did, iTerm 2 again eclipses Terminal by adding:
screen, and which most people regard as better & faster than
and a lot more. Some are mentioned here but some are not, such as co-processes, triggers,smart selection, semantic history, and so on. Development is pretty active, but documentation seems to lag behind. I highly recommend it. I've been using it for years now and have never missed Terminal.
(It's possible Terminal does some of the things I mention here--it's been so long since I've used it that I don't recall, but when I switched I paid close attention to the differences and there were lots of advantages to iTerm. And it keeps getting better every few weeks or months.)
You could also take a look as ZOC6 seems pretty cool.
Configure ssh options and connect to ssh hosts manually always make me feel bored and have a lot of trouble. At Codinn we made a couple of GUI tools to make using ssh effortless. Those tools helped ourselves, and may also help you.
Core Shell is a PuTTY alike tool with lots of extra features:
ssh_configfile as the source of advanced options, especially helpful for experienced users.
I'm getting along well with Royal TSX.
This is useful for SSH, RDP and VNC based terminals or web-based interfaces. It has a built-in credential management and team-sharing features.
You need to try Termius (available also as mobile app)
Fugu is what you are looking for.
If you want to start a gui program from ssh, you can use x11 and relay it with xeyes.
CyberDuck is a great option. I used it this semester in complement with Terminal. (CyberDuck is fully functional, we just coded in VI, so using the Terminal for SSH worked better for me.)
You can get CyberDuck for free online, or at a cost on the App Store.
Another option is FileZilla. I used it on Windows for FTP, but I believe that it supports SSH as well. It definitely runs on Mac as well as Windows, so it's another option. FileZilla is also free.