No, you had better not to go to the beach with pink eye. Firstly, pink eye is caused by bacteria and virus. Although there has quite low possibility to pass pink eye to others on the beach, pink eye is contagious. And you need to use eye drops to treat pink eye in an effective way, thus you had better not go to outside to make the pink eye worse. At the same time, the conjunctivitis can make your eye more sensitive to bright light until it's completely healed. What is more, the ocean contains salty water that would irritate your pink eye to make it worse.
It will not make your conjunctivitis worse, but you should take some measures to protect your eyes. Because the conjunctives can make your eyes more sensitive to the light, if your illness has not cured. So you can take a good pair of sunglasses. And, you cannot get sand in your eyes, wear a sun hat and sunblock, pink eye and a sunburn would look so funny.
You have no choice but to stay at home. Although pink eye is very common and is not serious, it will last 4-7 days without medical treatment and sometimes even longer for poor rest. Viruses or bacteria and dry eyes from lack of tears or exposure to wind and sun can cause pink eye. So staying at home and taking a note from a doctor and taking some medicines, then you'll recover soon.