Corrosion around battery cable terminals can be quite common. Cable terminals and battery posts should be cleaned about once a year.
At a minimum you will need to clean the terminals and battery posts and it’s possible that your car might need a new battery. If the battery is 5 or more years old and sits idle a lot then it’s questionable.
Battery cost will vary but a Protege should use a lower end, more inexspenive battery; probably in the 60-75 dollar range.
I think auto parts stores such as AutoZone will install one for you free and even Wal Mart may do the same.
They will also test the battery for you free of charge.