FTDI Vinculum code read out protection
How you approach this depends, of course, on who your adversary is and how valuable the information would be to them. Is your adversary the Mossad, or your neighbor's kid? Storing nuclear launch codes clearly requires a different level of protection than storing the password for your lawn sprinkler automation system :)

Cutting the programming pins will only make the task slightly harder; even a moderately-determined attacker could etch away the plastic and connect to the pins. I realize cost is an issue here, but you may consider putting the code on a microcontroller that does support readout protection, which will protect you against a casual adversary, but of course not against a well-funded one.

The problem of how to protect secrets (keys etc.) on hardware devices that are not always under the owner's physical control is a very large topic, with solutions ranging in price over several orders of magnitude.  Perhaps if you gave some more detail I could point you to the right information, but as it stands, all I can say really is "it depends".