There are many places in Spain and Portugal that have nice beaches as well as beautiful historical sites to visit. Fly to LIS and enjoy the beaches in nearby towns such as Cascais and Oeiras, for example.

I also really loved Croatia and Malta in terms of the combination of both beaches and lots of culture (Malta is better in the culture/historic sites aspect than croatia) but both are probably too hard to get to from NYC. Very easy from anywhere in Europe though.

And of course the obvious Rio de Janeiro. Classic, and of course rio has its downside (criminals) but it's still a great city with great beaches, both in the city and around it (a week trip to Búzios or Cabo Frio for example).