There are a number of ethnic spots that are BYO and places like Apiary and, I believe, Ciano that have special BYO nights (usually Monday).
For everyday BYO, La Sirene in Soho is one of the nicer spots and might make sense for a date. The food is good, traditional French bistro, and the setting is the same. It isn't anything cutting edge or amazing, and is a little cramped, but if BYO is important, it is a good choice.
Kuma Inn is the other spot that comes to mind, as it is a step up from some of the ethnic spots in terms of atmosphere. Not 100% certain they are still BYO, but always were.
113 Ludlow St, New York, NY 10002
558 Broome Street, New York, NY 10013
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45 E 22nd St, New York, NY 10010