Will a steering damper eliminate all tankslappers?  Are there any downsides to having a steering damper?

No, and $.

It helps prevent or mitigate the damage a tankslapper could produce, they are not cheap. Bike feels more comfortable lots of places now, such as rough pavement, etc.

One more suspension piece to maintain over the ownership of your bike.

They will do best with headshake - which can lead to a slapper. But if you've done something to initiate a full blown tank slapper, about the only way out is more throttle and less weight on the front.. of course giving more throttle to a bike that's tring to toss you off is not so easy.