PuTTY is a 32-bit application from the ground up, so it won't run on Windows 3.1 as a native 16-bit program; and it would be very hard to port it to do so, because of Windows 3.1's vile memory allocation mechanisms.
However, it is possible in theory to compile the existing PuTTY source in such a way that it will run under Win32s (an extension to Windows 3.1 to let you run 32-bit programs). In order to do this you'll need the right kind of C compiler - modern versions of Visual C at least have stopped being backwards compatible to Win32s. Also, the last time we tried this it didn't work very well.
If you're interested in running PuTTY under Windows 3.1, help and testing in this area would be very welcome!