An example implementation of a serializable Dictionary<string, int>

StringIntDictionary.cs

using UnityEngine;

namespace System.Collections.Generic
{
	[Serializable]
	public class StringIntTuple : SerializableKeyValuePair<string, int>
	{
		public StringIntTuple(string item1, int item2) : base(item1, item2) { }
	}

	[Serializable]
	public class StringIntDictionary : SerializableDictionary<string, int>
	{
		[SerializeField] private List<StringIntTuple> _pairs = new List<StringIntTuple>();

		protected override List<SerializableKeyValuePair<string, int>> _keyValuePairs
		{
			get
			{
				var list = new List<SerializableKeyValuePair<string, int>>();
				foreach (var pair in _pairs)
				{
					list.Add(new SerializableKeyValuePair<string, int>(pair.Key, pair.Value));
				}
				return list;
			}

			set
			{
				_pairs.Clear();
				foreach (var kvp in value)
				{
					_pairs.Add(new StringIntTuple(kvp.Key, kvp.Value));
				}
			}
		}
	}
}