Can I have a glass of wine or a beer with my picnic lunch in Central Park?

No. NYC has open containerlaws and alcohol is not permitted in public parks. If you are caught you can be fined around $100.

That being said, everyone does it. When there are free concerts in the parks, everyone brings wine and openly drinks it.
I wouldn't worry too much about being caught and fined.

If you're picnicking at one of the classical music concerts--go ahead with the wine.
Any other time, you do risk getting ticketed. Beer is bigger no-no. Any any alcohol at pop/rock/rap events is much more likely to be confiscated/
Fair? No, but that's the reality

Lots of people do it - at certain times. If it's one of the nights the philharmonic is playing a lot of people will have picnics with whine - but they wil be discreet.

Ifyou just sitting on a bench in the middle of the park and very overt about having an open bottle you may well get attention from the police since it is illegal. If they will enforce it depends on the cirumstances and people involved. But don;t assume you will get away with it.

NYC aint Paris. Do not drink alcohol anywhere outdoors, parks, beaches, yards, etc. Yes, people have gotten tickets and arrested for drinking outdoors in their own backyards and porches. The fuzz loves to ticket tourists, as they know they wont waste time with courts, hearings, complaints, IA, etc.
Watch out for plain-clothes cops.

The cops will use it against you if they want. Just tell them you have a cousin on the job in the seven five.