I assume this is a web form served via browser, yes?
For a simple one-time sent form I can't really see why anyone would ever need the information to be saved unless the sign up also functions as the login (given there's no password, I assume that doesn't apply). When in doubt, just disable the autocomplete if you value privacy over smooth re-entry. And in this case there seems to be no re-entry anyway.
Also do note that I'm not aware of any modern browser that saves autocomplete information unless you actually submit the form, so failing to complete the form wouldn't save any info anyway, unless you're talking some implementation you roll your own (you should generally not replicate browser features unless extremely necessary).
If the field is extremely private, like, say, credit card info or SSN, then I would say Autocomplete should be off entirely (though browsers have taken to requiring the security code when autocompleting credit cards).
Regarding public computers, I'm not sure this is a particularly useful case to worry about unless you have an unusual case where this is extremely common. Browser autocomplete asks people if they want to save sign-ins anyway and I would hope most libraries/etc simply link directly to the in-private versions of browsers these days. This is more something for the user and computer provider to worry about than it is for you unless, as above, it's ridiculously secure information you shouldn't save in any context.