For Westchester standards Glenn Island is a good public beach. They have good dressing rooms with big clean shower cabins, hot water lockers etc. In my opinion Beach in Rye by the Playland has prettier landscape, and It has a restaurant with outdoor seating just on the beach. Another interesting feature is that the sand in the tidal belt seems to be compressed - as if they have used a giant roller to do so. Pertinent effect keeps shallow waters clear by preventing sand to rise. On the other hand I am not sure if it has showering cabins and lockers abundant and spacious as those in New Rochelle.
If you are not familiar with Long Island Sound, take note that the waters are very shallow and that lamda Low-Hi tide is 10-20 meters ~ 12-24 yards. I suggest using New Rochelle tide chart on Google - fairly accurate and optimally spending time on beach and Long Island Sound Parks - Davenport, Hudson, Five Islands, Glen Island when the tide is High, if you want to be on optimistic a prettier side
There are many beach clubs in New Rochelle, check out Surf and others between Davenport Park and Glen Island.