How to call an explicitly implemented interface-method on the base class

Unfortunately, it isn't possible.
Not even with a helper method. The helper method has the same problems as your second attempt: this is of type B, even in the base class and will call the implementation of M in B:

interface I
{
  int M();
}
class A : I
{
  int I.M() { return 1; }
  protected int CallM() { return (this as I).M(); }
}
class B : A, I
{
  int I.M() { return CallM(); }
}

The only workaround would be a helper method in A that is used in A's implementation of M:

interface I
{
  int M();
}
class A : I
{
  int I.M() { return CallM(); }
  protected int CallM() { return 1; }
}
class B : A, I
{
  int I.M() { return CallM(); }
}

But you would need to provide a method like this also for B if there will be a class C : B, I...

It is possible using reflection.
The code follows. I added caching as a basic optimization, but it can be optimized further by using Delegate.CreateDelegate on methodInfo. Also, parameter count and type checks can be added using methodInfo.GetParameters().

interface I   
{   
    int M();   
} 

class A : I   
{   
    int I.M() { return 1; }   
} 

class B : A, I   
{   
    BaseClassExplicitInterfaceInvoker<B> invoker = new BaseClassExplicitInterfaceInvoker<B>();
    int I.M() { return invoker.Invoke<int>(this, "M") + 2; }   
}

public class BaseClassExplicitInterfaceInvoker<T>
{
    private Dictionary<string, MethodInfo> cache = new Dictionary<string, MethodInfo>();
    private Type baseType = typeof(T).BaseType;

    private MethodInfo FindMethod(string methodName)
    {
        MethodInfo method = null;
        if (!cache.TryGetValue(methodName, out method))
        {
            var methods = baseType.GetMethods(BindingFlags.NonPublic | BindingFlags.Instance | BindingFlags.DeclaredOnly);

            foreach (var methodInfo in methods)
            {
                if (methodInfo.IsFinal && methodInfo.IsPrivate) //explicit interface implementation
                {
                    if (methodInfo.Name == methodName || methodInfo.Name.EndsWith("." + methodName))
                    {
                        method = methodInfo;
                        break;
                    }
                }
            }   

            cache.Add(methodName, method);
        }

        return method;
    }

    public RT Invoke<RT>(T obj, string methodName)
    {            
        MethodInfo method = FindMethod(methodName);
        return (RT)method.Invoke(obj, null);
    }

}   //public static class BaseClassExplicitInterfaceInvoker<T>

Here is the source of my inspiration.