I have a class that contains an enum property, and upon serializing the object using JavaScriptSerializer, my json result contains the integer valu
No there is no special attribute you can use. JavaScriptSerializer serializes enums to their numeric values and not their string representation. You would need to use custom serialization to serialize the enum as its name instead of numeric value.
If you can use JSON.Net instead of JavaScriptSerializer than see answer on this question provided by OmerBakhari: JSON.net covers this use case (via the attribute [JsonConverter(typeof(StringEnumConverter))]) and many others not handled by the built in .net serializers. Here is a link comparing features and functionalities of the serializers.