How to animate Opacity of a DropShadowEffect?

A couple of points to note

1) You need to be targeting an actual property of Border - You are in effect trying to target the value (DropShadowEffect) of the Effect property, not the property itself.

2) You need to sort the syntax of the PropertyPath.

Change your Storyboard.Target property to the following and you should be fine:

Storyboard.TargetProperty="(Effect).Opacity"

EDIT Working code as noted in comment:

<Border BorderThickness="10" Height="100" Width="100">
    <Border.BorderBrush>
        <SolidColorBrush Color="Red"></SolidColorBrush>
    </Border.BorderBrush>
    <Border.Style>
        <Style TargetType="Border">
            <Style.Resources>
                <Storyboard x:Key="GlowOn">
                    <DoubleAnimationUsingKeyFrames BeginTime="00:00:00" 
                          Storyboard.TargetProperty="(Effect).Opacity">
                        <SplineDoubleKeyFrame KeyTime="0:0:0.3" Value="1"/>
                    </DoubleAnimationUsingKeyFrames>
                </Storyboard>
                <Storyboard x:Key="GlowOff">
                    <DoubleAnimationUsingKeyFrames BeginTime="00:00:00" 
                          Storyboard.TargetProperty="(Effect).Opacity">
                        <SplineDoubleKeyFrame KeyTime="0:0:0.3" Value="0"/>
                    </DoubleAnimationUsingKeyFrames>
                </Storyboard>
            </Style.Resources>

            <Setter Property="CornerRadius" Value="6,1,6,1" />
        <!--<Setter Property="BorderBrush"
                    Value="{StaticResource SelectedBorder}" />-->
            <Setter Property="BorderThickness" Value="1" />
        <!--<Setter Property="Background"
                    Value="{StaticResource DeselectedBackground}" />-->
            <Setter Property="RenderTransformOrigin" Value="0.5,0.5" />
        <!--<Setter Property="TextBlock.Foreground"
                    Value="{StaticResource SelectedForeground}" />-->

            <Setter Property="RenderTransform">
                <Setter.Value>
                    <RotateTransform Angle="90"/>
                </Setter.Value>
            </Setter>

            <Setter Property="Effect">
                <Setter.Value>
                    <DropShadowEffect ShadowDepth="20"
                                      BlurRadius="8"
                                      Color="#FFB0E9EF"/>
                </Setter.Value>
            </Setter>

            <Style.Triggers>
                <Trigger Property="IsMouseOver" Value="True">

                    <Trigger.EnterActions>
                        <BeginStoryboard
                              Storyboard="{StaticResource GlowOn}"/>
                    </Trigger.EnterActions>

                    <Trigger.ExitActions>
                        <BeginStoryboard
                              Storyboard="{StaticResource GlowOff}"/>
                    </Trigger.ExitActions>

                </Trigger>
            </Style.Triggers>

        </Style>
    </Border.Style>
</Border>