Yes, kind of. The scale will show something, but ultimately it will not be any downward force of your finger because that is not needed. If you apply a downward force then your finger would be accelerating downward.
When you first dip your finger into the water your finger will push down on the water a little and this force will be transmitted to the scale. However, this is a transient force arising from the inertia and viscosity of the water. Imagine slapping the water with an open hand. The impact that makes your hand hurt must also be transmitted to the scale.
Back to dipping your finger: When the water has moved out of the way and your finger is suspended motionless in the water, It will feel somewhat weightless to you (buoyancy) because your finger's density is almost identical to the density of water. The weight of the submerged portion of your finger will be transmitted to the scale. Imagine in the extreme case that the container is large enough for you to jump in all the way. The water container's weight will increase by your weight.
If you want to consider the difference in density, or are dipping something of vastly different density, then the more accurate way to look at it is that the force registered on the scale will increase by the weight of the water that is displaced by the submerged portion of the object, regardless of its density. Here are two examples:
1. Imagine forcing a helium balloon under water. 343
You have to push down to keep it under water. Your downward pressure will register on the scale even though the balloon is lighter than air. The downward pressure needed will match the weight of the water that is displaced by the balloon. (The balloon will also shrink as it is compressed by water pressure. The pressure and shrinkage increase with depth, so it will be a little easier to hold it deep than to hold it shallow.)
2. Now imagine dipping a heavy metal sinker into the water and holding it up with a string so that it does not touch the bottom of the container. The force you feel on the string will be a little lighter when the sinker is submerged, and the amount it will be lighter by will match the weight of the displaced water. This is because the sinker provides its own downward force to keep it submerged, so that portion of its weight will no longer be transferred up to you via the string. The scale will register only the addition of the weight of displaced water. The rest of the weight of the sinker will be carried by you via the string and will not show on the scale. (If the string is cut and the sinker drops to the bottom of the container, then the scale will show the entire weight of the sinker regardless of density.) 436