两个都不妥,问题的关键并不在 at .
1.这样的问句,道地的说法不用 what 而用 which.
2.do 不合适；通常只有在过去时问句中可能出现用did,现在时多半用进行式 are you staying,未来时可以用 will 或者 would.
最顺口的提问方式如下：
过去：Which hotel did you stay at?
现在：Which hotel are you staying at?
未来：At which hotel would you stay?或者 Which hotel will you be staying at?
不信,找个老外或华侨问问.