DNA computers? I think the answer is “we outclassed them”. At 10nm, transistors are about the size of 10 base-pairs at this point, but much faster and much less error prone.
Just isn’t something in super high demand. That and density are probably the only major advantages right? And I don’t imagine the equipment to "read" that data will be very compact in the foreseeable future.
I would imagine reading would be done in a similar way as it is today: amplification and sequencing. Data would have to be given a hefty amount of data correction. There are no apparent show stoppers that I’m aware of, just not enough interest.
Going with the Kuzweil estimate on Wikipedia, human brains are about 1e15 ops/Joule, whereas the best in the Green500 list (June 2019) is 1.76e10 ops/Joule.