Valentino - private alcoves - recommended?
Valenino's is always excellent. Hope you aren't driving because you will want to enjoy their amazing wine list. Don't worry if you don't know wine -- they will guide you. If you don't want to splurge on an expensive bottle, they have some excellent wines by the glass.
The alcoves can be romantic, particularly if you like getting a little frisky during dinner. (They are not THAT private, so nothing that you would write up in Penthouse Letters if thats what your thinking.) Otherwise I prefer sitting outside.
This is a classy place and you should expect to pay for it. I never feel ripped off, but I certainly prefer to go when someone else is paying!
I can vouch for the restaurant, but have never dined in the alcove, so I can't help with that. Excellent Italian, go for the tasting menu, and beautiful wine list. Fabulous birthday dinner choice.
Are you seeking for outdoor dining? If so, there are also other numerous options, like Lucques, Whist, Il Cielo, Geoffrey's...but Valentino, food-wise, is a top choice.