There is nothing better than a public course with a great view. Indian Island Country Club Golf Course, Riverhead, NY is one of my favorites. Only 20 minutes from my sister’s house in Westhampton, NY. Don’t let Country Club in the name fool you, this is a public course. Suffolk County residents get a great rate with a Green Key Card, $26 resident and $44 non-resident. 
The front 9 runs along Peconic River offering beautiful water views of boats and swans. Indian Island is a great course to walk. Forward tees are 5445, slope71.3, and rating 118. The Outer Banks Restaurant has some of the very best bar food in the area, including a great hamburger. On Tuesdays you get a free draft beer with purchase of a burger. Yum!
Montauk Downs State Park Golf Course is certainly more famous as Long Island public courses go, but the price isn’t worth the drive. 
$91 plus $20 for cart, I just don’t think it is worth the price. The course is not exceptional, and certainly not worth the money. It is a long course for woman 5797 yardage, slope 132 and rating 74.2. The Regulation yardage is 6289, slope 130, rating 71.2.
I’ll stick with playing Indian Island. I love the walk spoiled.
