“Beverage Menu” would be better English. Beverage is a modifier, so it’s not usually plural in this usage, at least not in common American usage.
Either sounds equally correct to me. There is nothing wrong with plural modifiers per se (do you say 'drink cabinet' rather than 'drinks cabinet'?); I think it just boils down to what one's used to hearing.
The only time I've ever (previously) heard of a "drinks cabinet" was when I was deliberately trying to search for noun adjunct expressions with a plural noun (on the notion that perhaps they existed somewhere, but I couldn't come up with any off the top of my head), and it came up as something said in the UK. And I've never heard of a "drink cabinet" till now (I take it from your comment that this might not be surprising; perhaps nobody anywhere uses it). In any case I'm not sure what this thing is. If I guess the meaning correctly-ish, an American equivalent might be "liquor cabinet," although I get the feeling "drinks cabinet" is a much more fixed, well-established concept.
Drinks cabinet is used in Australia. As is drinks menu..in a restaurant it would be standard for a waiter to say would you like to see the drinks menu.
I doubt too many Americans have ever uttered the phrase "beverages menu"; it's usually "drink menu" (I suppose I can imagine such a thing in a formal-enough establishment being labeled "beverage menu," but I can hardly imagine uttering the phrase otherwise). To me the plural noun adjunct thing is a strong indicator of non-American usage.
"Drink menu" is accepted, anyway, so I would assume they got "drinks menu," too.