Another thought: Look at Metrolink, the commuter train service that runs in LA, OC, Ventura, San Bernardino, and Riverside counties. The trains into/out of Downtown LA are pretty constant thru the day and into the night. You'd end up at Union Station, which is awesome. Across the street is Olvera Street, and Chinatown a short walk. Also, from Union Station, you can pick up the different subway routes, which would take you to Hollywood and Vine and Universal Studios. The Gold Line goes to Pasadena. As other posters recommended, you could stay in the Anaheim/D-Land area, San Dimas/Bonelli Park, or even the Riverside area and you would be near a train station. (There should be ART- Anaheim Resort Transit- shuttle service from the Anaheim RV park to ARCTIC, the new transportation center.)
Just remembered: I think Metrolink has a special $10 fare for all weekend.