With all the smaller older hotels in LA, it's hard to be certain, but when the Clark Hotel reopens - see Bringham Yen's post here, it might be the oldest building built as a hotel and still operating as a conventional hotel in Downtown Los Angeles. Or maybe even anywhere in Los Angeles.
The Pico House was built in 1869/1870, Plaza House in 1883, the Van Nuys/Barclay in 1896, the Hayward in 1905, the King Edward, the Alexandria & the Bristol in 1906, the Baltimore in 1910 and the Rosslyn in 1913 but it's been a long time since they've been hotels. The only other possible contender would be the hotel that is within the LA Athletic Club - assuming it opened concurrently with the rest of the building in 1912. And counting them all up - that means LA Downtown LA has at least eleven old hotel buildings that are at least a hundred years old - and likely a lot more