I'm not entirely sure why you'd want to do this. Here's a rephrase of your question which I'll use as the basis of my answer:
How can I launch a program on my computer from a webpage, passing information from the host as command-line arguments?
- Set up local server and use AJAX
- Server exec's commands from the client
- The game Mabinogi does this to launch the game from their site
- Use Flash or Java- They're plugins, so they execute outside of the browser sandbox
- Set up SSH tunnel between server and your computer
- Might have trouble getting a GUI application to run
This approach has some drawbacks, like you have to have the server installed for the page to work, but overall I think it's the best option.
However, if you just want a simple way to access remote servers without having to open putty first, I'm sure you could make a few launchers using putty's command-line parameters. Search Google for specific parameters to pass it.