Converting to upper is best as it covers compatible syntax for the 3 most-used Rails database backends. PostgreSQL, MySQL and SQLite all support this syntax. It has the (minor) drawback that you have to uppercase your search string in your application or in your conditions string, making it a bit uglier, but I think the compatibility you gain makes it worthwile.
Both MySQL and SQLite3 have a case-insensitive LIKE operator. Only PostgreSQL has a case-sensitive LIKE operator and a PostgreSQL-specific (per the manual) ILIKE operator for case-insensitive searches. You might specify ILIKE insead of LIKE in your conditions on the Rails application, but be aware that the application will cease to work under MySQL or SQLite.
A third option might be to check which database engine you're using and modify the search string accordingly. This might be better done by hacking into / monkeypatching ActiveRecord's connection adapters and have the PostgreSQL adapter modify the query string to substitute "LIKE" for "ILIKE" prior to query execution. This solution is however the most convoluted and in light of easier ways like uppercasing both terms, I think this is not worh the effort (although you'd get plenty of brownie points for doing it this way).