I have an older 60amp fuse box in a rental property that is wired as a sub panel off my main 100amp service. I would like to upgrade to a new circuit breaker panel so tenants can not pull out the 15amp fuse and up size it. The only box I can find at my local hardware store is rated 100amp (main lug and main breaker). If I replace the 100amp breaker in the box I found with a 60amp breaker would that be fine? I assume that the box is rated up to 100amp but anything lower would work too.
You will also likely need to replace the cable with larger gauge wires, though without knowing what's there right now it's impossible to say for sure. Also, whenever you replace old electrical equipment, you are generally required to adhere to the latest code, so it's possible there are other changes you will need to make too. This might include AFCI breakers.
If you are unsure, always best to check with a licensed electrician.
Yes, "down-rating" panels by installing a lower amperage main breaker of a type that meets the panel's listing is fine -- in fact, you can often order the exact same panelboard from the factory with different main breaker amperage options, or main lugs -- and then change it out freely in the field depending on installation conditions.
Electrically, that's fine. I have no idea whether code would object.